THE SCIENCE 


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MATHEMATICS LISRARY 


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THE SCIENCE 


OF 


0 mm 


AND 
Their * Practical * Application 
FOR 


TEACHERS AND PRIVATE LEARNERS 


BY 


MERRITT L. DAWKINS 


DOWNING, MO. 
PEERLESS PUBLISHING COMPANY. 
1890. 


Entered according to act of Congress, in the year 1890, by 
MERRITT L. DAWKINS, 
In the Office of the Librarian of Congress at Washington. 


; PRINTED AT 
THE RECORD PRINTING HOUSE, 
DOWNING, MO. 


ISMy 44 Spoes evel. Quer, 


513 
paza MATHEMATICS LIBRARY 


I NTRODUCTORY. 


———— 


The design of the author in preparing this little 
volume has been to embrace, within moderate space, a 
practical and useful treatise on ARITHMETICAL SCIENCE, 
and to place it within the reach of every home and every 
business office throughout the land. 


The author does not arrogate to himself the first 
knowledge of the methods found in this work. He sim- 
ply contends that no similar work issued previous to the 
date of his copyright is so convenient, instructive and 
satisfactory. 

By the utmost brevity and precision the author 
has been enabled to compress a vast number of 
useful methods into a small compass and by eliminating 
a few subjects which serve rather to perplex than to en- 
Jighten, he has, in the opinion of those competent to 
judge, produced a work that will prove valuable to the 
business man, the farmer, the mechanic, and the large 
class of people who have not had the opportunity or 
ability to master the various technical terms of arithmetic, 
and to whom many of its principles and processes have 
heretofore been laborious and difficult. 


The author has availed himself of many valuable hints 


and suggestions from business men, practical teachers, 


6022183 


4 INTRODUCTORY. 


and educators, all of whom he desires to thank most 
cordially for the aid they have rendered. 

The greatest care has been taken to ensure accuracy, 
but where imperfection is so general in works of this 
class, it is too much to hope that errors have not crept 
in. Should the reader note any such, the author will 
deem it a favor to be informed of them, with a view to 
their correction in subsequent editions. 

Trusting that the work will in some measure supply 
the popular demand for a Practican AritHmertic, the 
author presents his work to the public. 

M.: dae; 


September 1, 1890. 


NOTICE. 


We solicit correspondence with teachers and others 
who are open for a profitable engagement. 

This is by universal consent the best selling book pub- 
lished, and if you want to make Bia Money secure an 
agency for this great work without delay. 

Sample copy mailed on receipt of one dollar. 

PEERLESS PUBLISHING Co. 


SGIENGE OF NUMBERS. 


~~ ——— 


DEFINITIONS. 


1. A Number is a unit or a collection of units. 


2. An Even Number is one that is exactly divis- 


ible by 2. 


3. An Odd Number is one that is not exactly divis- 
ible by 2. 

4. A Prime Number is one that has no exact di- 
visor besides itself and 1. 


5. A Composite Number is one that has exact 
divisors besides itself and 1. 


6. An Abstract Number is one that denotes no 
particular thing. 


@. A Denominate Number is one which denotes 
some particular thing; as two pounds, four boys. 


$. An Integer is a whole number. 


9. The Complement of a number is the difference 
between it and a unit of the next higher order. 


10. The Supplement of a number is the difference 
between it and a unit of the next lower order. 


11. The Reciprocal of a number is 1 divided by 
that number. 


ADDITION. 


>< 


2. Addition is uniting two or more numbers into 
one. 


IS. The Addemnds are the numbers to be added. 
if. The Sumi is the result obtained by adding. 


15. The Sign of Addition is the perpendicular 
cross, +, and is read Pius. 


16. The Sign of Equality consists of two hori- 
zontal parallels, = , and is read EQUALS, Or, IS EQUAL TO. 


iv. WRapidity and acuracy in addition can be se- 
cured only by frequent and careful practice. These two 
acquirements are the most Sse qualifications of an 
accountant. 


18. Careful Practice will enable the student to 
add several columns at once. This method cannot well 
be extended to more than three columns with any prac- 
tical advantage, unless the columns are incomplete. 


19. The Work should be tested by adding in an 
opposite direction from that in which the additions were 
_first made. 

20. When there are but Two Numbers to be 
added, it is more convenient to begin at the left to add, 


observing to carry ONE when the sum of the next lower 
figures is more than 9. 


ADDITION. 7 


21. Make Combinations of tens when possible 
and think resutts only. Thus, instead of saying 3 and 
“ai 7 and 8 are 15, think 8, 7, 15. 


22. Numbers increasing by a common difference, 
as 2,4, 6, 8, may be added by multiplying the first and 
last by the number of addends. 

23. Write Totals of each column under each other, 
by one of the two methods shown below, and then com- 
- bine them in one result. The first method is perhaps 
the better, as it obviates the difficulty of carrying tens. 


24. 1. A has 798 dollars, B 875 dollars, and C 769 


dollars. How much have they together? $2442. 
PROCESS. ANOTHER PROCESS. 
22 798 2'2 
22 875 2/4 
22 769 24 
2442 


2. I owe one man $375, another. $280, a third $564, 
a fourth $119, a fifth $75. How much do I owe? 

7 $1413. 

3. Paid for coffee $245, for tea $325, for sugar $196, 
for flour $217, and for spices $273. What did all cost? 

$1256. 

4, <A drover paid $265 for one horse, $198 for another, 
$237 for another, and $216 for another. What did he pay 
for all? #916. 

5. Four men formed a partnership. A furnished 
$2467, B $1674, C $2312, and D $1893. What was their 
capital? $8346. 

6. A farmer has 328 sheep in one field, 675 in an- 
other, 682 in another, 729 in another, and 2380 in another. 
How many has he in the five fields? 4794, 
7. How many strokes does a clock strike in 12 
hours? 78, 


he) ADDITION. 


8. A carpenter receives $246 for one job, for another 
$327, for another $683, for another $456, and for another 
$713. How much did he receive for all? $2425. 

9. A merchant pays $320 a year for rent, $1230 to one 
elerk, $832 to another, and $438 for sundry expenses. 
What does his business cost him a year? $2820. 

10. A landlord receives $329 rent for one house, $671 
for another, $476 for another, $524 for another, $387 for 
another. How much does he receive for all? $2387. 

11. A gentleman bought a store for $625, and paid 
$148 for repairs, and $175 for having it enlarged. For 
how much must he sell it, in order to gain $160? 


$1108. 
12. A man owns six horses. The first is worth $384, 


the second $488, the third $348, the fourth $843, the 
fifth $483, and the last $834. What is the value of the 
six horses? $3330. 
13. A gentleman having lost $2380 finds that he has 
$7289 remaining. How much would he have had, if in- 
stead of losing, he had found $2380, and then earned 
$2256 more? $14305. 
14. Having a note due, I paid $153 at one time, $216 
at another, $435 at another, $673 at another, $327 at an- 
other, and there were $174 still unpaid. What was the 
face of the note? $1978. 
15. A gentleman making his will, left $3260 to his 
wife,-$2875 to his oldest son, $1890 to his second son, 
$1250 to his youngest son. What amount did he be- 
queath in his will? $9275. 
16. A man sold his piano for $409, his collection of 
paintings for $541, his library for $718, his carpets for 
$726, other furniture for $1738. How much money did 
he obtain by the sale? $4182. 
17. A drover bought horses for $628, and cows for 
$382; on his horses he gained $324, and on his cows $125. 
What should he have received for both to have gained 
$216 more than he did? $1675. 


ADDITION. Q 


18. What is the sum of all numbers from 1 to 24 in- 
clusive? 300. 

19. A boy agreed to work 30 days for 2 cents the 
first day and an increase of 2 cents per day. What 


did he receive? $9.30. 
20. What is the sum of all the odd numbers from 1 
to 49 inelusive? 625. 


21. In one book there are 785 pages, in another 867 
pages, in another 879 pages, and in another 684 pages. 
How many pages in the four books? 3215. 

22. A merchant pays $98 a month to his clerks, $60 
to his bookkeeper, $37 to his janitor, $18. for fuel, $11 
for gas, $36 for rent, and $70 for advertising. What 
are his expenses per month? $330. 

23. There are 31766 square miles in Maine, 9280 in 
New Hampshire, 10212 in Vermont, 7800 in Massachu- 
setts, 1306 in Rhode Island, and 4674 in Connecticut. 
How many square miles in all of New England? 

65038. 

24. The population of the five principal cities of 
Missouri in 1880 was as follows: St. Louis 350222, 
Kansas City 55813, St. Joseph 32484, Hannibal 11074, 
Sedalia 8202. What was their entire population at’ that 
time? 457795. 

25. Jackson, Greenslate & Co. ground 11936 pounds 
of wheat on Monday, 10117 pounds on Tuesday, 8135 
pounds on Wednesday, 9963 pounds on Thursday, 7215 
pounds on Friday, 12634 pounds on Saturday. How 
many pounds did they grind during the week? 

60000. 

26. A gentleman being asked his age, said he was 18 
years old when he Jeft the common school, he spent 3 
years in college, 4 years in a medical school, practiced 
his profession 24 years, was a member of congress 8 
years, and it was 13 years since he retired from business. 
How.old was he? 70. 


SUBTRACTION. 


a 


25. Subtraction is taking one number from an- 
other. 


26. The Minuend is the number from which an- 
other is to be taken. 


26. The Subtrahend is the number to be sub- 
tracted. 


24@. The Remainder is ths result obtained by sub- 
tracting. 


28. The Sign of Subtraction is —, and is read 
MINUS. 


29. Only Like Numbers can be added or sub- 
tracted. 


30. It is Generally more convenient to write the 
subtrahends under the minuend, but sometimes it is 
better to reverse this position. The student should be 
able to subtract either downward or upward as may be 
required. 

3i. Test the Work by adding the remainder to 


the subtrahend. If their sum is equal to the minuend, 
the work is correct. 


32. Several Numbers may be subtracted from 
another at a SINGLE OPERATION, as shown in the following 
example. 


SUBTRACTION. 1] 


33. 1. Ina regiment of 1280 men, 248 are killed in 
battle, 324 are taken prisoners, 192 are drowned, and 273 
join the enemy. How many remain? 243, 


PROCESS. 
1280 


248 
324 
192 
273 


243 

Expianation.— Add each column of subtrahends sep- 
arately, and write in the remainder such figure as added 
to the sum of each column will make its right hand figure 
equal to the corresponding figure of the-minuend. The 
sum of the first column is 17 and we perceive that 3 
must be added to it to make 20, the next higher number 
in which the unit figure in both terms are ALIKE, hence 
we write 3 in the remainder and carry 2 to the next 
column, the sum of which is 24 which, taken from 28 
leaves 4 in the remainder; carry 2 and proceed as be- 
fore. 

2. A merchant has 3050 dollars. What sum will he 


have after paying 2364 dollars? $686. 
3. A farm that cost 8275 dollars was sold at a loss of 

396. What was the selling price? $7879. 
4. Bought a farm for 3280 dollars, paying 1820 dol- 

lars down. How much remained unpaid? $2460. 


5. J. Morgan & Co. owe me on account 8350 dollars. 
If they pay 6428, how much remain unpaid? $1922. . 
6. Boughta farm for 1820 dollars and sold it for 
2460 dollars. How much did I make by the bargain? 
$640. 
7.° If a man’s income is 900 dollars a year and his 
expenses 678 dollars, how much can he save in a year? 


$222. 


ii SUBTRACTION. 


8. From a tract of land containing 10000 acres the 
owner sold to A 4750 acres, and to B 875 acres. How 
many acres had he left? 4375, 

9. If I have $2000 on deposit, and give a check for 
#135 to one man, to another for $673.50, and to another 
for $429.50, how much will remain? $762. 

10. The captain of a ship having a cargo worth 16375 
dollars, threw overboard in a storm 7532 dollars worth, 
What was the value of the goods left? $8843. 

11. Charles had 376 cents, James gave him 432 cents, 
and Henry gave him enough to make his number 1000. 
How many cents did Henry give him? 192. 

12. A merchant had a quantity of flour on hand, for 
which he asked 780 dollars, but for cash he sold it for 36 
dollars less. How much did he receive for his flour? 

$744. 

13. A merchant having 248 pounds of coffee, bought 
198 pounds, and then sold 73 pounds to one man, and 
184 pounds to another. How many pounds had he left? 

189. 

14. A man set out on a journey of 1250 miles. Dur- 
ing the first day he traveled 342 miles, and during the 
second day he traveled 416 miles. How many miles had 
he yet to travel? 492. 

15. A merchant worth 20000 dollars, lost a store by 
fire worth 2750 dollars and goods to the amount of 3650 
dollars. How much had he left after receiving 3200 
dollars insurance? $16800. 

16. A man having 6789 dollars, invested 500 dollars 
in bank stock, 480 dollars in a creamery 675 dollars in a 
mining stock, and 1200 dollars in United States bonds. 
How much had he left? $3934. 

17. A man owns property valued at 6480 dollars, of 
which 2316 dollars are in personal property, and 1673 
dollars in real estate. The remainder is deposited in 
bank. How much has he in bank? ; $2491. 


x 


SUBTRACTION. 13 


18. A gentleman had 4800 dollars to destribute among 
his three sons. 4’o the eldest he gave 1834 dollars, to 
the second 1683 dollars, and the remainder to the young- 
est. How much did the youngest receive? $1283, 

19. If a man’s annual income is 3600 dollars, and he 
spends 363 dollars for house rent, 267 dollars for cloth- 
ing, 684 dollars for provisions, 416 dollars for servants, 
and 238 dollars for traveling, how much will he have left 
at the end of the year? $1632. 

20. On Monday morning a bank had on hand 1826 
dollars. - During the day 2191 dollars were deposited 
and 3412 drawn out, on Tuesday 3256 dollars were de- 
posited and 2164 dollars drawn out. How many dollars 
were on hand Wednesday? $1697. 

21. At the end of the year 1890, [ found I had spent 
2400 dollars. Of this amount 350 dollars were paid for 
board, 125 dollars for clothing, 475 dollars for books, 
150 dollars for incidentals, and remainder for real estate. 
what was the cost of the real estate? $1300. 

22. The following is my private account for two weeks: 
First week, received 75 dollars as salary and spent 27 
dollars for clothing, 6 dollars for board, 4 dollars for 
washing, and 7 dollars for sundries. Second week, re- 
ceived 75 as salary, loaned 25 dollars to A. D. Lewis, 
paid 6 dollars for board, 3 dollars for washing, and 16 
dollars for sundries. How much did I have at the end 
of the two weeks? HHK. a, WY 


MULTIPLICATION. 


$4. Multiplication is taking one number as many 
times as there are units in another. 


3. The Multiplicamd is the number to be multi- 
plied. 


36. The Multiplier is the number by which we 
multiply. 


3¢@. The Product is the result obtained by multi- 
plying. 

38. The Sign of Multiplication is the oblique 
cross, X, and is read TIMES. 


39. The multiplicand and multiplier are called the 
Factors of the product. 


49. Ordinary Multiplication is so well known 
that it requires no explanation. The methods given in 
this work are quite practical and very easily learned. 
No one method of contraction is best for all cases, but it 
is well for the pupil to be familiar with several and apply 
the most appropriate in each particular case, or any one 
that seems most simple may be adopted for general use. 


441. A portion of the Multiplication Table is 
inserted here for the benefit of those who are not 
thoroughly acquainted with it. Now, dear reader, if you 


MULTIPLICATION. 15 


belong to this class, do not rest until you have mastered 
PERFECTLY that portion of the table here given. 


TABLE. 


5 6 7 8 91011 12 
6| 8} 10) 12) 14| 16] 18) 20| 22) 241 


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72 


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| | 63, 72, 81) 90 99 108 
110 120 
121132 
182)144 


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NUMBERS LESS THAN 20. 


43. The following method will be found valuable in 
Multiplying Mentally any two numbers less than 20. 


44. 1. Multiply 16 by 14. 224. 
é PROCESS. 


16 
14 


200 
24 


224 


Expranation.—Add the unit figure of one number to 


16 MULTIPLICATION. 


the other number, annex a cipher, and to the result add 
the product of the unit figures. After a little practice 
the operation can be performed mentally, thus extending 
the multiplication table to the twenties. 


2. How many miles will a man travel in 16 days, if 


he travel 18 miles per day? 288. 

3. If there are 17 yards of cloth in one piece, how 
many yards in 13 pieces? 221. 

4. Ifaman can dig 19 bushels of potatoes in one 
day, how many bushel can he dig in 16 days? 304. 

5. Ifa boat can sail 18 miles per hour, how many 
miles can she sail in 14 hours? 252. 


EN dS DIN Gra VWVils ee CP EL Bi: 


45. To Multiply when there are ciphers at the 
right of one or both factors, proceed as if there were no 
ciphers, then annex the ciphers to the result. 


46. 1. Multiply 3200 by 60. 192000. 
PROCESS. 
3200 
60 
192000 
2. What will 42 cows cost at 40 dollars a head? 
$1680. 
3. What costs 56 yards of muslin at 30 cents a yard? 
$16.80. 
4. What will 21 horses cost at the rate of 80 dollars 
per head? $1680. 
5° What will be the cost of 70 acres of land at 24 
dollars per acre? $1680. 


6. Ifa ship sail 28 miles an hour, how many miles 
will she sail in 60 hours? 1680. 


MULTIPLICATION. 17 


SMALL NUMBERS. 


47. The Method of multiplying by 11 and other 
small numbers is explained below. 


48. 1. Multiply 236 by 11. 2596. 
PROCESS. 
2596 


ExpLanation :— Write the unit figure and then add the 
digits, two at a time, beginning at the right, finally writ- 
ing the left hand figure. 


2. Multiply 423 by 16. 6768. 
PROCESS, 


423 
16 


6768 


Expianation:—Multiply through by 6 and carry the 
succeeding figure of the multiplicand each time, finally 
writing the left hand figure of the multiplicand increased 
by the units to carry from the next lower order. 


3. Multiply 123 by 24. 2952. 
PROCESS. 


123 
24 


2952 
ExpLaNaTion:—Proceed as above, but carry double 
the succeeding figure each time, or double the multipli- 
cand mentally and multiply by half the multiplier. 


4. Ifa clerk receive 125 dollars a month, how much 
will he receive in 16 months? $2000. 


18 MULTIPLICATION. 


5. If it takes 132 laborers 18 months to build a rail- 
road, how many months will it take 1 man to build it? 


2376. 
6. Allowing 365 days to the year, how many days has 
a man lived who is 24 years old? 8760. 
7. Sound moves 1142 feet in a second, how many feet 
will it move in 27 seconds? 30834. 
8. A cooper can make 127 barrels in a week. How 
many can he make in 17 weeks? 2159. 


9. Ifaman put 28 dollars in a savings-bank in a 
month, how much will he dposit in 14 months? $392. 


FACTORING. 


49. The ordinary method of factoring the multiplier 
when it is a Composite Number requires no ex- 
planation. 


50. To Multiply when one part of the multiplier 
is a factor of another part, the following contraction will 


be found valuable. 
51. 1. Multiply 421 by 312. 131352. 


PROCESS. 


421 
312 


1263 
3052 


131352 
Expianation.—Multiply first by the 3, and then this 
result by 4, writing each partial product in its proper 
order. 


2. If an ocean steamer sail 287 miles per day, how 
many miles will she sail in 84 days? 24108. 


MULTIPLICATION. 19 

3. Sold my farm of 248 acres at 186 dollars per acre. 
How much did I get for it? M6128. 

4. How much will it cost to lay 325 miles of ocean 
cable at an expense of 217 dollars per mile? = $70525. 

5. What will be the cost of 239 bushels of potatoes at 
84 cents per bushel? $200.76. 


MULTIPLIER NEAR 100, 1000, ELC. 


52. The method of Multiplying by a number a 
little less than a unit of the next-highest order is illus- 
trated below. 

53. Ll Multiply 135 by 98. 13230. 

PROCESS. 
13500 
270 


13230 


ExpLaNnation.—Annex two ciphers to the multiplicand, 
thus multiplying by 100, and from the result subtract 
the product of the multiplicand by the complement of 
the multiplier. 


2. What will be the cost of 216 bushels of corn at 


97 cents per bushel? $209.52. 
3. What is the value of 796 pounds of tea at 96 
cents per pound? ) $764.16. 
4, What will 175 acres of land cost at 97 dollars per 
acre? $16975. 
5. What is the value of 1234 bushels of wheat at 95 
cents per bushel? $1172.30. 


6. A nurseryman counted the trees in his orchard and 
found that he had 127 rows, each row containing 980 
trees. How many trees were in the orchard? 124460 


20 MULTIPLICATION. 


ALCIQOUGT PARTS; 


54. Much Time can be saved when the multiplier 
is an aliquot part of some higher unit, by abbreviating 
the ordinary method as here illustrated. 


55. 1. How many trees in 327 rows containing 25 
trees to the row? 8175. 


PROCESS. 
4)32700 
S175 


Expianation:—Multiply by 100 by annexing two 
ciphers, and then divide by 4. 


TABLE: 

Of 10 Of 100 Of 1000 
14 is, 4:64 1s 4y5)1624---is- ae 
185 ris kt 8k is eye O68 Shige ss 
y iss $y LOS RISP Wey tooe ess ae 
24 is 4,123 is 4/125 is #32 
3 is 4/162 is +4 | 1662 is + 
5 is #¢/]20 is 4/250 is 4 
62 is 2/25 is 4/ 3334 is 34 
7% - is» 44334658 4 | 750 is 4 
Bhar id 5S 00 isan ae | Os pe de ee 


2. What will be the cost of 28 yards of cloth at 123 
cents per yard? $3.50, 


MULTIPLICATION. 21 


3. Find the cost of 256 yards of cloth at $1.064 per 


yard. $272. 
4. What will be the cost of 258 cords of wood at 
$3.334 per cord? $860. 
5. What will 176 bushels of corn cost at 374 cents per 
bushel? $66. 
6. What will 192 bushels of wheat cost at $1.163 per 
bushel? $224. 
7. What will be the cost of 48 yards of silk at $1.163 
per yard? $56. 
8. What is the value of 225 barrels of flour at $3.332 
per barrel? $750. 
9. What will 368 bushels of potatoes cost at 624 cents 
per bushel? $230. 
10. What will be the cost of 324 pounds of tea at 75 
cents per pound? $243. 
11. What is the value of 216 bushels of apples at 
$1.374 per bushel? $297. 
12. What will be the cost of 72 gallons of wine at 
$1.125 per gallon? $81. 
13. When butter is worth 334 cents per pound, what 
will 786 pounds be worth? $262. 


NUMBERS ENDING WITH 5. 


54. The following method of multiplying any two 
numbers whose Unit Figures are 5 will be found 
valuable. 


55. 1. Multiply 45 by 25. 1125. 
PROCESS. 


45 
25 


1125 
ExpLanaTIon :—Multiply the figures in tens place to- 


22 MULTIPLICATION. 


gether, increase this by half their sum, and to the result 
annex 25, or if the sum of the figures in tens place is 
odd, annex @5. 
2. What cost 75 acres of land at 35 dollars per acre? 
$2625. 
3. How many bushels of corn can be raised on 65 
acres of land at the rate of 35 bushels per acre? 


2275. 
4, If 35 men can build a wall in 25 days, how many 
days will it take one man to build it? 875. 


5. Ifit requires 125 tons of iron rail for one mile of 

railroad, how many tons will be required for 45 miles? 
| 5625, 

6. A merchant bought 115 yards of cloth at 35 cents 
per yard. What did it cost? $40.25. 

7. What will be the cost of 85 pounds of butter at 


45 cents per pound? $38.25. 


NEWTON § MULTIPLICATION RULE. 


38. This Valuable Method may be used in mul- 
tiplying any two numbers in which the unit figures add 
to TEN and the. other figures are ALIKE. 


49. 1. Multiply 26 by 24. 624, 
PROCESS. 
26 
24 
624 


ExpLANAtion:—We say 4 times 6 are 24, put down 
both figures and carry 1 to the second figure of the mul- 
tiplier. Then say 3 times 2 are 6. Always carry ONE 
to the tens figure of the multiplier. The product of the 


MULTIPLICATION. 23 


unit figures must occupy two places, hence, if their 
product is less than TEN, a cipher should be written in 
the product. 


2. If aman travel 33 miles per day, how many miles 
will he travel in 28 days? 924. 


PROCESS. 


33 
28 


924 


Note:—The above rule is likewise applicable when. 
the digits of the multiplicand are auixe and the digits of 
the multiplier add to TEN, also when the complement of 
the unit figure in the multiplier cross-multiplied by the 
tens in the multiplicand is equal to the product of the 
other figures cross-multiplied. 


3. What will 84 bushels of wheat cost at 86 cents per 
bushel? $72.24. 

4. How many square rods in a field 147 rods long and 
48 rods wide? 7056. 

5. How man pounds of sugar are there in 126 packages, 
each package weighing 86 pounds. 10836. 

6. How many bushels of wheat can be raised on 39 
acres of land, if one acre produces 24 bushels? 936. 

7. What will be the cost of 77 bushels of corn at 37 
cents per bushel? $28.49. 

8. What cost 64 bushels, of apples at 38 cents a 
bushel? $24.32. 

9. A travels 30 miles a day, and B travels 38 miles a 
day. How many miles will both travel in 36 days? 


2448, 
10. How many sheets of paper in 26 quires, if there 
are 24 sheets in a quire? 624. 


11. A mining company built 395 tenement houses at 


24 MULTIPLICATION. 


an average cost of 395 dollars apiece. What did they all 


cost? : $156025. 
12. A drover bought 42 oxen at 384 dollars a head. 
What did they cost? $1617. 
13. What cost 29 pounds of coffee at 21 cents per 
pound? $6.09. 
13. What is the value of 1574 yards of cloth at 48 
cents per yard? £75.60. 


COMPEE MENU EMU DASE Ish ART Giarss 


38. This rule is valuable when both multiplicand 
and multiplier are a Little Less than a unit of the 
next higher order. 


39. 1. Multiply 96 by 94. 9024. 
PROCESS. 
96-4 
94 -6 
90 24 
ExpLanation:—Multiply the complements of both fac- 
tors together and write the result in the product. Then 
cross-subtract; that is, take the complement of one num- 
ber from the other number, and write the remainder at 
the left of the first product. If the product of the com- 
plements does not contain as many figures as the multi- 
plier, insert cipher to make up the deficit. 


2. At 97 dollars per acre, what will a farm of 96 acres 


cost? $9312. 
3. What will be the cost of 89 bushels of corn at 92 
cents per bushel? $81.88. 


4. Ifa planet travel 996 miles per minute, how far 
will it travel in 996 minutes? 99216. 


MULTIPLICATION. 25 


5. What will 92 horses cost at 97 dollars per head? 


$8924. 
6 How many square feet in a garden 93 feet long and 
86 feet wide? 7998. 
7. What will be the cost of 196 yards of cloth at 95 
cents a yard? $186.20. 
8. Ifa vessel sail 93 miles a day, how far will she 
sail in 98 days? 8649. 
9. What will be the cost of 92 bushels of rye at 95 
cents a bushel? $8740. 
10. A has 99 dollars, and B has 61 times as much. 
How much has B? $6039. 
11. How much will I receive for a farm of 95 acres at 
48 dollars per acre? | $4560. 
12. What must I pay for 990 town lots at an average 
price of 994 dollars? #98460. 
13. How many square rods in a square field which 
measures 92 rods on each side? 8464. 


14. <A farmer sold 94 bushels of wheat at 92 cents a 
bushel. What did he get for it? $86.48. 


15. A drover bought 96 head of cattle at 85 dollars 
per head. What did they cost him? $8 160. 


16. <A grocer packed 92 cases of eggs, each containing 
8 dozen. How many eggs did he pack? 8832. 


17. Mercury moves in its orbit at the rate of 975 miles 
per minute. How many miles will it movein 975 miutes? 
’ 950625. 


18. A travels 37 miles a day and B travels 42 miles a 
day. How many miles will they both travel in 96 days? 
iw 
(584, 


19. The earth is 95 million miles from the sun, and 
Neptune is 42 times as far. How many million miles is 
Neptune from the sun? 3990. 


26 MULTIPLICATION. 


17. Two ships start from the same place and sail in 
opposite directions, both at the rate of 48 miles a day. 
How many miles apart will they be in 91 days? 8736. 


GROSS VEU ie Tt bale ites erie 


60. This Valuable Method now for the first time 
presented to the public in its simplest form, has been 
known perhaps as long as the ordinary method, but the 
wise men of past ages guarded the beautiful system with 
so much care and secrecy that it has been kept out of 
school and college text books up to the present day, and 
hence is not generally known. This rare and practical 
method, soon to become the common property of man- 
kind, is now given in its complete and unabridged form, 
as a free offering to the world. 


61. Cross Multiplication appears very complh- 
cated, but is quite easy, and a little practice will enable 
the student to multiply rapidly and accurately by a 
number consisting of several figures. This method 
should be adopted for general use by all who desire to 
compute with rapidity. 


62. The Ordinary Method may be used for test- 
ing the work until the pupil becomes familiar with the 


process. » 
63. 1. Multiply 54 by 32. 1728. 
PROCESS. 
54 
32 


1728 


Nw 
~] 


MULTIPLICATION. 
® 


EXxpLaANATion.—-First. 2 times 4 are 8. 
Second. 2 times 5 + 3 times 4 are 22: write 2 and 


carry 2. 
Third. 3 times 5 + 2 are 17. 
2. Multiply 256 by &4. 21504. 
PROCESS. 
256 
st 
21504 


ExpLaNnation:— First. + times 6 are 2-44: write 4 and 
carry 2. 

Second. - times 5 + 8 times 6 + 2 are 70; write O 
and carry @. 

Third. 4 times 2+ 8 times5 + 7 are 55:3 write 5 
and carry 5. 

Fourth. 8 times 2+ 5 are 21. 


3. Multiply 649 by 325° 210925, 
PROCESS. 
649 
325 
210925 


EXpLANATION:—First. 5 times 9 are 45; write 5 and 
carry 4. 

Second. 5 times 4+ 2 times 9 + 4 are 42: write 2 
and carry 4. 

Third. 5 times 6+ 3 times 9 + 2 times 44+ 4 are 
69; write 9 and carry 6. 

Fourth. 2 times 6 + 3 times 4 + 6 are 30; write O 
and earry 3. 

Fifth. $3 times 6 -! 3 are 21. 


28 MULTIPLICATION. 


4, “Multiply 1357 by 2468. 3349076. 
PROCESS. 
1357 
2468 
3349076 


Exp.Lanation:—First. 8 times 7 are 563 write 6 and 
carry 3. 

Second. 8 times 5 + 6 times 7 + 5 are 87: write @ 
and carry 8. 

Third. S& times 3 + 4 times 7 + 6 timesS + 8 are 
90; write 0 and carry 9. 

Fourth. Stimes i+ 2 times @ + 6 times 3+ 4 
times 5 + 9 are 69; write 9 and carry 6. 

Fifth. 6 times I + 2 times 5 + 4 times 3 + 6 are 
34; write £4 and carry 3. 

Sixth. 4 times Il + 2 times 3+ 3 are 13; write 3 
and carry 1. 

Seventh. 2 times i + I are 3. 


5. What cost 169 yards of silk at $3.25 per yard? 
$549.25. 
6 What must I pay for a farm of 360 acres at 27 dollars 
per acre? $9720. 
7. At the rate of 53 miles per hour, how far will a 
car run in 54 hours? 2862. 
8. How much can a man earn in 64 months at a salary 
of 56 dollars a month? $3584, 
9. What will be the cost a tract of land containing 
4528 acres at 27 dollars per acre? $122256. 
10. A is worth 5482 dollars, and B has 134 times as 
much. How much have they both? $740070. 
11. One cord of wood. contains 128 cubic feet. How 
many cubic feet are there in 75 cords? 9600 


12. A travels 79 miles a day and B 40 miles. How 
much farther does A travel in 31 days than B? — 1209. 


EE ————— ee ee 


MULTIPLICATION. 29 


13. A regiment of soldiers has provisions to last 718 
men 32days. How many days would they last one man? 
22816. 
14. J. Morgan & Co. make 2528 handles per day. At 
this rate how many handles can they make in a year of 
312 days? 788736. 
15. It requires 2176 pickets to fence a square lot. 
How many pickets will be required to fence 1575 lots of 
the same size and shape? 3427200. 
16. A drover sold 260 sheep at $3.20 per head, and 
425 hogs at an average price of $9.64 per head. How 
much did he get for them? $4929, 
17. A drover bought 1386 cows at 382 dollars each, 
and the same number of horses at 123 dollars each. 
\\ hat did he pay for both? $21080. 
18. A farmer counted the trees in his orchard and 
found that he had 324 rows, each row containing 536 
trees. How many trees were there in the orchard? 
173664. 
19. Two vessels are 4320 miles apart and sail towards 
each other, one at the rate of 26 miles per hour, the other 
at the rate of 18 miles per hour. How far apart will 
they be in 37 hours? 2692. 
20. Mr. Chas. Hale sold a farm of 325 acres at $65.50 
per acre, and received in payment 345 sheep at $3.25 per 
head, a note for $2684.95, and the rest in cash. How 
much cash did he receive? $17481.30. 
21. Two ships sail from the same place, in 
opposite directions, one at the rate of 127 miles per day, 
and the other at the rate of 119 miles per day. How far 
will they be apart in 49 days? 12054. 


DIVISION. 


>< 


64. Division is finding how many times one num- 
ber contains another. 


65. The Dividend is the number to be divided. 
66. The Divisor is the number by which we divide. 


67. The Quotient is the result obtained by divis- 
ion. 


68. The Remainder is the number which is some- 
times left after dividing. 


69. The Sign of Division is +, and is read DIvID-— 
ED BY. Division is also indicated by writing the dividend 
above or at the right of the divisor, with a line between 
them. 


70. Short Division is that in wich the steps in 
the solution are performed mentally. 


@i. Long Division is the method of dividing 
when all the work is written. 


@2. When a Remainder occurs at the end of divis- 
ion, it may be written over the divisor in the form of a 
fraction and annexed to the quotient. 


DIVISION. ; 31 

73. When the Divisor is large, consider only the 

first two or three figures as a trial divisor, and compare 
them with the first two or three figures of dividend. 


@4. To Test the work, multiply the divisor by the 
quotient, and to the product add the remainder, if any. 
If the work is correct, the result will equal the dividend. 


FACTORING: 


@5. Frequently, when the divisor can be resolved in- 
to Factors, the work can be shortened by dividing by 
each of the factors in succession. If there is more than 
one remainder, the true remainder may be found by sub- 
tracting the product of the divisor and quotient from the 
dividend. 


@6. 1. Find the quotient of 594 divided by 18. 
33. 


Expianation :— Divide the dividend by one factor of 
the divisor, the quotient by another, and so on tillall the 
factors are used. 


2. In one hogshead there are 63 gallons. How many 


hogsheads in 15435 gallons? 245. 

3. Ifa boat sail 24 miles an hour, how many hours 
will it be in sailing 1824 miles? 16, * 
4. The product of two numbers is 26973; one of the 

numbers is 81; what is the other? aon. 


5. A farmer raised 8288 bushels of wheat, averaging 
56 bushels to the acre. How many acres did he plant. 
148. 


3:2 DIVISION. 

6. A man bought 160 acres of land at 25 dollars an 
acre, giving in payment a house valued at 928 dollars, 
and horses at 96 dollars each. How many horses did he 
cive? 32. 


CR NG Es TIN Gee Deere 


@¢. This method often renders it quite easy to apply 
Short Divisiom to large numbers when there are 
ciphers at the right of the divisor. 

@%. 1. Divide 8765 by 600. 14; Rem. 365. 

PROCESS. 
6 00)87 | 65 
14 - 385 

ExpLanation:—Cut off the ciphers at the right of the 
divisor, and the same number of places at the right of 
the dividend. Divide the remaining part of the dividend 


by the remaining part of the divisor, and prefix the re- 
mainder to the figures cut off for the true remainder. 


2. If 120 acres of land cost 4080 dollars, what will 


one acre cost? $34. 
3. A drover paid 28800 dollars for 360 horses. How 
much was that per head? $80. 
4, Aman gave 5600 dollars for 160 acres of land. 
How much was that per acre? $35, 
5. How many months musta person labor to earn 
3430 dollars at 70 dollars per month? 49, 
- 6. Into how many lots of 40 acres each can a tract of 
land containing 6400 acres be divided? 160. 


7. Two men start from the same place and travel in 
opposite directions, one at the rate of 23 miles a day, and 
the other at the rate 27 miles a day. When 4750 miles 
apart, how many days had they traveled? 95. 


a 


nf 


DIVISION. 33 


ALIQUOT PARTS. 


79. This method of division is the reverse of Multi- 
plication by aliquot parts, shown on page 20. 
$0. 1. Divide 450 by 25. | 18. 
PROCESS. 
456 
4 


18(00 


Expianation:—Multiply by 4 and then divide by 
100, by cutting off two places. ' 


2. At $1.25 per yard, how many yards of silk can be 


bought for 15 dollars? 12. 
3. A dealer bought 25 horses for 2650 dollars. How 
much did he give a head? $106. 
4. At 163 cents per yard, how many yards of muslin 
can be bought for 8 dollars? 48, 
5. At 123 dollars per barrel, how many barrels of flour 
can be bought for 450 dollars? 36. 
6. Ifaman can save 250 dollars per year, in how 
many years can he save 2250 dollars? J, 
If a locomotive run 25 miles an hour, how many 
hours will it be in running 1625 miles? 65. 
8. If it takes 163 yards of silk to make a dress, how 
many dresses can be made from 450 yards? eer 
9. How many weeks will it take a printer to earn 
250 dollars, if he receives 162 dollars a week? 15. 


10. A farmer received 14 dollars for a load of apples 

at 334 cents a bushel. How many bushels did he have? 
42. 

11. A farmer bought a tract of land for 2000 dollars. 

How many acres did he buy, if it cost him 123 dollars 
per acre? 160. 


. 34 DIVISION. 


12. If the distance across the Atlantic ocean is 3000 
miles, how many days will a vessel, sailing 125 miles per 
day, be in crossing? 24. 


COMPLEMENT DIVISION. 


$i. When the divisor is a Little Less than 100, 
1000, ete., the following method will be found valuable. 


$2. Divide 2138672 by 98. 21823; Rem. 18. 

PROCESS. 

21386 | 72 

427 | 72 

8 | 54 

| 16 - 

21823 | 14 

brekie’ 

is 


Expianation:—Cut off from the right of the dividend, 
by a vertical line, as many figures as the divisor contains. 
Multiply the part on the left of the line by the comple- 
ment of the divisor and set the product under the divi- 
dend. Multiply the part of this product on the left of 
the line by the same multiplier and set down as_ before. 
Continue in like manner until no figure remains on the left 
of the line. Add theseveral results and forevery 1 carried 
across the line, add the number used as a multiplier. 
The part on the left of the line will be the quotient, 
and the part on the right, the remainder. Ifthe remain- 
der is equal to or larger than the divisor, carry one to 
quotient, and the excess will be the true remainder. 
This rule may be advantageously applied to many other 
numbers. To illustrate: If we wish to divide by 49 
we may divide by 98 and multiply by 2, or if we wish 


er 


DIVISION. 35 
to divide by 198, we may divide first by 99 and then 
that result by 2. 


2. Divide 2326 by 32. 72; Rem. 22. 
PROCESS. 


23 | 26 
92 


ExpLanation:—In this case we divide by 96, or three 
times the given divisor, which gives a quotient of 24. 
This multiplied by 3 gives 72, which is the true quoti- 
ent. 


3. In one cask there are 94 gallons. How many casks. 


in 4230 gallons? 45, 

4. Ifaship sail 99 miles a day, in how many days 
will it sail 12375 miles? 125. 

5. In 240 quires of paper there are 5760 sheets. How 
many sheets in a quire? 24. 

6. In one day there are 24 hours; how many days 
are there in 5200 hours? 216 da. 16 hr. 

7. A man bought a drove of 95 horses for 4750 dollars. 
How much did he give apiece? $50. 

8. Amansolda farm of 480 acres for 15360 dol- 
lars. How much did he get per acre? $32. 

9. If aman can earn 998 dollars in a year, how many 
years will it take him to earn 15968 dollars? 16. 


10. A farmer having 6272 dollars bought land at 32 
dollars per acre. How many acres did he buy? 196. 

11. A man wishes to invest 1645 dollars in railroad 
stock. How many shares can he buy at 47 dollars per 
share? 35. 


CANCELLATION. 


>< 


$3. Cancellation is the method of shortening 
computations by omitting common factors from both 
dividend and divisor. 


$4. Write the dividend on the right and the divisor - 
on the left of a verticaL LINE. Cancel all common fac- 
tors and divide the product ofthe remaining factors of 
the dividend by the product of the remaining factors of 
the divisor. 


$5. <A Factor in one term will cancel Two or more 
factors in the other term, when their product is equal to 
the former. 


$6. Cancellation is frequently rendered easier by an- 
nexing Decimal Ciphers. For example: Suppose 
we have every number on the left of the line canceled, . 
except 4, it also will disappear by cancelling, if we annex 
two decimal ciphers to some number on the right. 


$@. In some cases it is better to reduce decimals to 
Common Fractions. To illustrate: If we have $1.50 
on the right of the line, perhaps it would be better to call 
it of a dollar, writing the 8 on the right of the line 
and the 2 on the left. 


$8. To Test the work, write the statement as it was 


PROPERTIES. 37. 


first written, with the result also on the left, and if all 
cancel, the work is correct. 


$9. 1. Divide 4 x 5 x 7 by 4x 7. D. 
STATEMENT. 


4,4 
5 
W\7 


2. How many barrels of flour, at $8 a barrel, must be 
given for 48 tons of coal, at 6 dollars a ton? 36. . 

3. How many yards of muslin, at 35 cents per yard, 
will pay for 20 yards of calico, at 14 cents per yard? 

8, 

4. How many cords of wood, at 4 dollars per cord, 
must be given for 3 tons of hay at 12 dollars per ton? 

9. 

5. How many bushels of corn, at 63 cents a bushel, 
must be given for 7 pieces of cloth, each containing 27 
yards, at 40 cents a yard? 120. 

6. A farmer bought 12 cows at 25 dollars a_ head, 
and paid for them in hay at 15 dollars per ton. How 
many tons of hay were required? 20. 

7. A farmer buys 4 pieces of cloth, each containing 
60 yards, at 15 cents per yard and pays for it with wheat 
at $1.50 per bushel. How many bushels are required? 

24. 

8. Exchanged 15 pieces of calico, each containing 
30 yards at 10 cents a yard, for 3 pieces of muslin, each 
containing 50 yards. What was the muslin per yard? 

$0.30. 

9. I sold 70 kegs of nails of 100 pounds each at 5 cents 
a pound, paying for them with pieces of calico of 35 
yards each, at 8 cents a yard. How many pieces did I 
give? 125. 

10. A merchant bought 24 tubs of butter, each contain- 
ing 32 pounds, at 40 cents a pound, paying for them with 


38 PROPERTIES. 


4 patterns of silk of 48 yards each. How much was the 
silk per yard? $1.60. 
11. A miller bought 4 loads of wheat, each containing 
18 sacks of 2 bushels each, worth $1.25 per bushel, and 
paid for it in flour at 6 dollars per barrel. How many 
barrels of flour were required? 30. 
12. I bought 75 barrels of apples, of 4 bushels each, at 
40 cents a bushel and paid for them with barrels of po- 
tatoes, of 5 bushels each, at 50 cents per bushel. How 
many barrels of potatoes did I give? 48, 
13. <A dealer exchanged 480 bushels of wheat worth 
70 cents a bushel, for an equal number of bushels of 
barley, worth 84 cents per bushel, and corn worth 56 
cents per bushel. How many bushels of each did he 
receive? 240. 


GREATEST COMMON DIVISOR. 


90. The Greatest Common Divisor of two or 
more numbers is the greatest number that will exactly 
divide each of them. 

91. 1. What is the greatest common divisor of 42, 
84, and 126? 42. 


PROCESS. 


alas - $4 - 126 


3/21 - 42 - 63 
CPi tr dene tS | 
CS eee og 


2x3x7=42 


ExpLANATION.—-Write the numbers one beside another, 


PROPERTIES. 39 


with a vertical line at the left, and divide by any com- 
mon factor of Att the numbers. Divide the quotient in 
like manner, and so on until the quotients have no com- 
mon factor; the product of all the divisors will be the 
greatest common divisor. 


2. A man bought three pieces of land containing 28, 
36, and 44 acres respectively, which he wished to divide 
into the largest possible fields, each having the same 
number of acres. How many acres can he put ina field? 

4. 


3. A gentleman having a triangular piece of land, 
the sides of which are 176 feet, 224 feet, and 320 feet, 
wishes to enclose it with a fence having panels of the 
greatest possible uniform length. What will be the 
length of each panel? 16. 

4, I have 3 pieces of carpet, the first containing 54, 
the second 90, and the third 126 yards. If these pieces 
were cut into the largest possible equal pieces, what will 
be their length? 18. 

5. A farmer wishes to put 182 bushels of wheat and 
234 bushels of oats into the least number of bins possi- 
ble, that shall contain the same number of bushels. How 
many bushels must each bin hold? 26. 

6. <A grocer had 126 apples and 189 pears which he 
wished to divide into smaller heaps, each containing the 
same number. What is the largest number that each 
heap may contain? 63. 


LEAST COMMON MULTIPLE. 


92. The Least Common Multiple of two or 
more numbers is the least number exactly divisible by 
them. 


40 PROPERTIES, 


93. 1. Find the least common multiple of 12, 30, 
and 70. 420). 


PROCESS. 


2/12 - 30 - 70 


S106 2 se eenanl 


Be LR BS 


Peay bee ie 
2 Xb Oe ed ee AeO 


Exprianation:— Write the numbers one beside another, 
with a vertical line at the left, and divide by any prime 
number which will exactly divide rwo or more of them. 
Write the quotient and undivided numbers beneath. Di- 
vide these in the same way, and thus continue until no 
two numbers in the lowest line have a common divisor. 
The product of these numbers and the divisors will be 
the least common multiple. A composite number may 
be used for a divisor, when it is contained in ALL the 
given numbers. 


2. How long must a box be that no room may be lost 
in packing in it books 6 inches, 8 inches or 12 inches 
long? 2 ft. 

3. The least common multiple of 12, 16, 21, and some 
unknown prime number to these three, is 3696. What is 
the unknown number? af 

4. How many bushels will the smallest bin contain 
that can be emptied by taking out either 3 bushels, 5 
bushels, or 14 bushels at a time? 210. 

5. What is the least sum of money for which I can 
purchase either sheep at 4 dollars; cows at 21 dollars, 
oxen at 49 dollars, or horses at 72 dollars a head? 

$3528. ° 


PROPERTIES. 41 


6. Whatis the smallest number that is exactly di- 
visible by 16, 24, and 30? 240. 

7. Two men start at the same time to walk around a 
field. One can go around in 3 hours, the other in 4 
hours. In how many hours will they again meet at the 
starting place? 12 


~_s 


HANDY HINTS AND HELPS. 


94. 'Twois an exact divisor of any even number. 


95. Three is an exact divisor of any number, the 
sum of whose digits is divisible by 3. 


96. Four is an exact divisor of a number, if the 
number expressed by its two right hand figures is divisible 
by 4. 


97. Five is an exact divisor of any number ending 
with O or 5. 


98. Six is an exact divisor of any EVEN number, 
the sun of whose digits is divisible by 3. 
99. Eight is an exact divisor of a number, if the 


‘number expressed by its three right hand figures is divi- 
sible by 8. 


109. Nine is an exact divisor of any number the 
sum of whose digits is divisible by 9. 

101. Ifan Kvea Number is divisible by an odd 
number, it is also divisible by twice that number. 

102. Any Number consisting of three equal digits, 
as 111, 222, ete., is divisible by 37, and the sum of the 
digits will be the quotient. 


103. The Product of any three consecutive numbers 
is divsible by 6. 


42 PROPERTIES. 

104. The Product of any four consecutive numbers 
is divisible by 24. 

105. The Produet of any four consecutive numbers 
plus 1 is a perfect square. 


106. Odd Numbers wultiplied together always 
give an odd number. 


107. The Geometrical Mean of two numbers is 
equal to the square root of their product. 


108. The Product of the sum and difference of 
of two numbers is equal tothe difference of their squares. 


109. If the Difference of two numbers be added 
to their sum, the amount will be double the larger num- 
ber. If their difference be subtracted from their sum, 
the remainder will be double the smaller number. 


= 


COMMON FRACTIONS. 


110. A Fraction is a part of a whole number. 


Hit. Common Fractions are expressed by two 
numbers, one written above the other with a horizontal 
line between them. 


112. The Numerator is the number which is writ- 
ten above the line. 

113. The Denominator is the number which is 
written below the line. 


114. A Proper Fraction is one in which the 
numerator is less than the denominator. | | 

1415. An Improper Fraction is one in which the 
numerator equals or exceeds the denominator. 


116. A Mixed Number is one consisting of an 
integer and a fraction. 

Li7. A Compound Fraction is a fraction of a 
fraction. 

118, A Complex Fraction is one having a_frac- 


tion in one or both its terms. 


119. To reduce fractions to Larger Terms, mul- 
tiply both terms by the same number. 


44 COMMON FRACTIONS. 


120. To reduce fractions to Smaller Termes. 
divide both terms by the same number. 


121. To reduce mixed numbers to Improper 
Fractions, multiply the whole number by the given 
denominator, to this product add the numerator, if any, 
and write the result over the given denominator. 


122. To reduce improper fractions to whole or 
Mixed Numbers, divide the numerator by the denom- 
inator. 


£23. To reduce two fractions to a Common BDe- 

nominator, multiply the denominators together for a 
common denominator and cross multiply for the numer- 
ators--that is multiply the numerator of the first fraction 
by the denominator of the second, and the numerator 
of the second by the denominator of the first. “ 


124. If there are several fractions it is better to find 
by inspection the Least Common Denominator, 
divide this denominator by the denominator of each fraec- 
tion and multiply both terms of the fraction by the 
quotient. 


125. To reduce Complex Fractions, write the 
product of the means for the numerator and the product 
of the extremes for the denominator of a simple fraction. 


126. To Add or Subtract fractions, reduce them 
to similar fractions, and add or subtract the numerators 
as required and place the result over the common denom- 
inator. When there are mixed numbers, add or sub- 
tract the whole numbers and the fractions separately. 


127. To Multiply fractions, multiply the numera- 
tors together for the numerator, and the denominators 
for the denominator of the product. Cancellation will be 
found valuable in this case. Write the numerators on 
the right of the line and the denominators on the left. 


COMMON FRACTIONS. 45 
128. To Multiply two fractional numbers ending 
with 4, take the product of the whole numbers, increase 
this by $ their sum and annex 4 to to the product. 
129. To Divide fractions. invert the divisor and 
multiply. 
130. Business Wen only care to have. the answer 
correct to the nearest cent that is they disregard the 
fraction. When it is a half cent or more they call it an- 


other cent. 


lL. Reduce ee ® to YAths. 8, 18 ae 
2. Reduce +2, +2 to lowest terms. 2 3 
3. Reduce 3, 3, to a common denominator. Br, te. 


4. Reduce 3, \4, 2 to their least common denominator. 


5 64 0 


603 60> 0. 
5. Five men share $11.56} equally. What is the 


share of each? $2.314. 
6. Change % to an equivalent fraction having 24 for 
its denominator. oy. 


7. Change #3 to an equivalent fraction expressed in 


its lowest terms. - 


8. Change 4% to an equivalent fraction, having 9 for 


its denominator. 3 
9. What will be the cost of 125 pounds of butter at 
124 cents per pound? $1.564. 


10. Ifa boy earn } of a dollar per day, how much 
can he earn in 12 days? $9. 

11. A man owned a boat worth 2700 dollars. How 
much should he receive for ? of it? $1800. 


12. A hunter shot 24 ducks one day, and } as many 
the next. How many did he shoot in both days? = 44. 


46 COMMON FRACTIONS. 
13. When wheat is selling at 3 of a dollar per bushel, 
how many bushels can be bought for 24 dollars? 30. 
l4. If from a bin containing 375% bushels of wheat, 
2163 bushels are taken, how many bushels will remain? 
159%. 
15. James spent 4 of a dollar on Monday and 4 of a 
dollar on Tuesday. What part of a dollar did he spend 
both days? 3. 
16. <A farmer received 21% dollars for hay, 343 dollars 
for a cow, and 98? dollars for a horse. How much did 
he get for all? $1555 
17. What will be the cost of 74 yards of muslin at 
at 12$ cents per yard, and 125 yards of gingham at 18} 
cents per yard? $3,285. 
18. What will be the cost of 8 loads of coal, each con- 


taining 15? bushels, at 124 cents per bushel? — $15.75. 


STATEMENT. 
8 

4 | 63 
2| 25 


19. A farmer sold ? of his cows, and ? of his sheep, 
and then had 270 of each remaining. How many of 
each had he at first? Cows, 450; Sheep, 945. 

20. A man walked 254 miles on Monday, 28} on 
Tuesday, 273 on Wednesday, 26% on Thursday. How 
many miles did he walk? ° 1083. 

21. A clerk earns $374 per month and pays $12} for 
board, $23 for washing, and $43 for other expenses. 


How much does he save? $173. 


COMMON FRACTIONS. 47 

22. A lady having a 20 dollar bill paid $34 for dress- 

voods, $9} for a bonnet, and #43 for sundries. How 
much money had she left? $2. 

23. A farmer received } of a dollar for eggs, 3 of a dol- 

lar for butter and £ of a dollar for other produce. How 
much did he receive for all? $234. 

24. By selling 8 yards of cloth for 24 dollars, | made 

* of the cost. TI paid for it with wheat at 3 of a dollar 
per bushel. How many bushels did I give? 25. 

25. James has ? of an orange. He gives Charles 3 

of this and divides the remainder equally between three 


boys. What part does each of the three boys receive? 
1 


Re 


DECIMALS. 


131. A Decimal is one or more of the decimal di- 
visions of a unit. 


132. A Pure Decimal is one which consists of 
decimal figures only. 


133. A Mixed Decimal is one which consists of 
« whole number and a decimal. 


134. A Complex Decimal is one which contains 
a common fraction at the right of the decimal. 


135. A Cireulating Decimal is one in which a 
figure or set of figures is continually repeated in the 
same order. 


136. Add and Subtract decimals as ordinary 
numbers. 


137. Multiply decimals as whole numbers, and in 
the product point off as many decimals as are in both 
factors. “4 


138. Divide decimals as in whole numbers, and 
point off in the quotient as many decimal places as the 
dividend has more than the divisor. Decimal ciphers 
may be annexed to the dividend if necessary. 


139. Common Fractions may be reduced to 
decimals by dividing the numerator by the denominator, 


DECIMALS. 49 


140. A Circulating Decimal may be reduced 
to a common fraction, by writing the repetend, or repeat- 
ing part, for the numerator of a fraction, with as many 
9s as there are places in the repetend for the denomi- 
nator. 


I41t. The various denominations of United States 
Money have decimal relations, and for ordinary busi- 
ness purposes two decimal places are sufficient. In writ- 
ing results, count fractions amounting to five mills or 
more as one cent and discard smaller fractions. 


142. All Contractions used with integers, may 
also be used with decimals. 


1435. 1. Reduce 13 toa common fraction. sy. 

2. What cost 33.21 yards of cloth at $4.41 per yard? 
$146.46. 

3. There are 7.92 inches in a link. How many inches 
in 300 links? 2376. 

4. When shingles are worth $2.50 per thousand, how 
much will 3750 cost? $9.38. 

5. A merchant sold 31.25 yards of muslin for $7.8125. 
How much was that per yard? $0.25. 

6. When land is worth $137.18 per acre, how much 
must be paid for a farm of 38 acres? $5212.84. 


7. Cowell & Co. sell galvanized wire at $3.75 per hun- 
dred. How much must I pay fora bale containing 118 
pounds? $4.24. 

8. Bought 3 loads of hay, the first containing 1.06 
tons, the second 1.05 tons, and the third 1.25tons. What 
did it cost at $5.12 per ton? $17.22. 

9. I bought 32) bushels of wheat at the rate of 16 
bushels for $10.04, and sold it at the rate of 20 bushels 
for $17.50. What was my profit? $79.20. 

10. A dealer bought 3 loads of wood, the first contain- 


ing 1.07 cords, the second 1 cord, and the third .945 
cords. What did it cost at $3.60 per cord? $10.85. 


50 DECIMALS. 
11. A grain dealer expended $70.15 in the purchase of 
rye at $.95, wheat at $1.37, and corn at $.73 per bushel, 


buying an equal quantity of each. How many bushels 
did he purchase in all? 69. 


CONTRACTED MU td) rishi Aas 


144. When the Product of two decimal numbers 
is not required to contain figures below a certain order, 
the work may be shortened as illustrated below. 


145. 1. Multiply 26.48352 by 12.452, retaining two 


decimals in the result. 329.77 
ORDINARY METHOD. CONTRACTED METHOD. 
2648352 26.48352 
12.452 254.2 1 
51|296704 2648 4 
132 \ 41760 a297 
1059;'\)3408 1059 
5296 :170A 132 
26483 52 > 
$329.7¢7\' 279104 29.04 


Expianation:—In the contracted method the order of 
the figures of the multiplier is ReEveRseD and the unit 
figure written under the decimal of the multiplicand to 
be retained. Begin with the right hand figure of the 
multiplier to form partial products, dropping one figure 
of the multiplicand, as we multiply by each successive 
figure of the multiplier, making due allowance for the 
units arising from the product of the neglected figures, 
adding the nearest number of tens. Ciphers may be 
annexed to the multiplicand if necessary. 


2. Multiply 123.215263 by 15.16, reserving two deci- 
mals in the product. 1867.94 


DECIMALS. D1 


CONTRACTED DIVISION. 


146. Contracted Division of decimals is the re- 
verse of contracted multiplication. 
147. 1 Divide 329.7727915 by 26.48352, retaining 
two decimals. 12.45. 
PROCESS. 


2648 352)3829.77279 15 (12.45 
264858 


649 


EXPLANATION: —In this case we see by inspection 
that the first two figures of the quotient will be whole 
numbers, and that four divisions must be made. Us- 
ing as many figures of the divisor as there are divisions 
to be made, we multiply by the first figure of the quo- 
tient, making due allowance for units arising from the 
product of it and the last omitted figure of the divisor. 
At each successive division omit one figure of the divisor, 
beginning at the right. 


2. Divide 1867.94361727 by 123.215263, reserving two 
decimal places in the quotient. 15.16. 


iy. OF ILL LIB. 


PERCENTAGE. 


f4%. The Base is the number on which percentage 
is computed. 


149. The Rate is the number which denotes how 
many hundredths are to be taken. 


150. The Percentage is the product obtained by 
multiplying the base by the rate. 


151. The Amount is the sum of the base and _per- 
centage. 


152. The Difference is the difference between the 
base and percentage. 

153. The Sign of Percentage is 4, and is read 
PER CENT. 


154. The following Equations are adapted to the 
solution of all simple problems in percentage. 


FORMULAS. 
Percentage = Base x Rate. 
Rate = Percentage + Base. 
Base = Percentage + Rate. 
Amount = Base x (1 + Rate.) 
Difference = Base x (1 — Rate.) 


PERCENTAGE. 53 


155. 1. A farmer who hada flock of 270 sheep, 
sold 334% of them. How many did he sell? 9). 


PROCESS. 
3)270 
90 


Exputanation: Take sucha part of the given nuim- 
ber as the rate per cent is of 100. 


2, What per cent is lost by selling cloth at 75 cents 


which cost 1 dollar? 25 
3. What per cent is gained by selling cloth for 1 dol- 
lar which cost 75 cents? 334 
4. A farmer having a flock of 250 sheep sold 20¢ 
of them. How many had he left? 200. 
5. Bought corn at 50 cents a bushel. At what price 
must it be sold to gain 20 per cent.? $.60. 


6. A farmer sold a horse for 75 dollars, which was at 
a loss of 25%. What was the cost of the horse? $100. 
7. Bought cloth at 50 cents per yard and sold it at 
60 cents per yard. What per cent did I gain? 20 
8. A man sold a horse at a profit of 334%. If the 
horse cost him 120 dollars, what did he get for him? 
$160. 
9, A farmer sold a horse for 100 dollars, which was at 
a gain of 257. What was the cost of the horse? 
SSO. 
10. [send my agent 500 dollars to invest in goods. 
After deducting 3% commission how much will be invest- 
ed? $485. 
ll. By selling cloth at $4.50 per yard I lose 104. 
What will be my gain per cent, if I sell at 6 dollars per 
yard? 20. 


INTEREST. 


>< — 


156. Interest is money charged for the use of 
money. 

157. The Principal is the sum on which interest 
is counted. 

158. The Amount is the sum of the principal 
and interest. 

159. Simple Interest is counted on the prin- 
cipal only. 

160. Compound Interest is interest on both 
principal and unpaid interest. 

161. The old Cancellation Method of comput- 
ing interest is undoubtedly the best for general use. The 
following diagram is intended to illustrate how the terms 
should be arranged. 


FORMULA. 


ee 


Principal. ' 


ExpLaNnation:—We write the principal, rate per cent, 


—- 4 @ 


INTEREST. Dy» 
and number of days to run, on the right, and 860, orits 
factors, on the left. If the time is in months, write #2 
only on the left. Point off two places in the result when 
the principal contains dollars only, and if it contain cents, 
point off two more. The operation may be shortened in 
various ways. Suppose the rate is 6 per cent, the result 
would be the same if we omit the rate from the right 
and write 60 only on the left. The divisor for any rate 
may be found by dividing 360 by the rate. In count- 
ing the time. regard 30 days as a month. 


162. 1. Find the simple interest on 120 dollars for 
1 month and 12 days, at &7. $1.12. 


STATEMENT. 


|1290 
360 O08 
| 42 
2. Find the simple interest on 600 dollars for 24 days 
at 7 per cent. $2.80. 
3. Find the simple interest on $36.84 for 5 months, at 
% per cent. #1.38. 
4. What is the simple interest on 640 dollars for 21 
days, at 6 per cent? $2.24. 
>). Find the simple interest on 720 dollars for 1 year, 
4 months, 15 days, at 6 per cent. $59.40. 
6. Find the simple interest on 2500 dollars for 7 
months and 20 days, at 5 per cent. $79.86. 
7. Find the simple interest on 6 dollars for 6 years, 
6 months and 6 days, at 6 per cent. $2.35. 
8. What will be the amount of 720 dollars for 20 
days at 6 per cent, simple interest? $722.40. 
9. Find the simple interest on 125 dollars for 1 year, 
2 months and 12 days, at 6 per cent. $9. 
10. Find the amount of $256.20 for 1 year, 3 months 
and 18 days, at 10 per cent, simple interest. $33.31. 


D6 INTEREST. 


11. Find the amount of 200 dollars, for 2 years, at 6 


per cent, compound interest, payable annually. 
$224.72. 


PARTIAL PAYMENTS: 


163. Partial Payments made on notes, mort- 
gages and other interest-bearing certificates of indebted- 
ness are called INDORSEMENTS. 


164. Nearly every Business Man has his own 
method of computing interest when partial payments 
have been made, and the subject has given rise to much 
litigation. We give only the method which has been 
adopted by the Supreme Court of the United States. 


UNITED: STATES METHOD: 


165. Find the Amount of the principal to the time 
of the first payment, and subtract the payment for a new 
principal. If the payment is less than the interest, 
find the amount of the note to the time when the sum of 
the payments shall exceed the interest due. Subtract 
the sum of the payments and proceed as before. 


166. This excellent Method is very simple. The 
only difculty occurs when the interest is greater than 
the payment. When this occurs, however, we can gener- 
ally see by inspection that the payment is not sufficient 
to discharge the interest,so that in such cases we 
may compute the amount at once up to the time when 
the sum paid is not less than the accrued interest, thus 
saving unnecessary work, by mentally estimating the in- 
terest in advance. 


INTEREST. 57 


167. 1. A note for 150 dollars is dated June 12, 
1888. Indorsed: October 12, 1889, 32 dollars; October 


What was 
$120.68. 


12, 1790, $6.80; February 12, 1891, $21.60. 


due May 27, 1892, interest at 6 per cent? 
PROCESS. 


Days. 
i2 
12 
12 
12 
24 


Years. Mon. 
ss - 6 - 
$9 10 
90 10 
91 - 2 - 


92 _ | ee 
ca ; Paid. 
$32.00 


It - 4 - 90 


tt : 0 - Oo - eae 
Oo - 4 - 0 21.90 


pa eae ek et ies (1° Pw) 


150.00 
12.00 


Principal. 
Ent. lt yr. 4 mon. 


162.00 
32.00 


Amount. 
Payment. 


130.00 New Principal. 


10.40 


140.40 
28.40 


112.00 
§.68 


Int, l yr. 4 mon. 


Amount, 
Payment. 


New Principal. 
Int. lL yr. 3 m. 15 d. 


Balance due. 


ExpLanation:—Write the dates under each other, 


in 


5S INTEREST. 


their regular order, subtract downward, and write the 
intervals beneath with the payments opposite. 

The first interval is 1 yr. 4 mon.; the interest on 150 
dollars for this time is 12 dollars and the amount is 
162 dollars; 162 dollars less 32 dollars is 130. dol- 
lars, the second principal. 

Consider mentally that the next interval of I year 
would afford about 8 dollars interest, which is more 
than the payment; hence we combine two intervals and 
their payments, making the time I yr. 4 mon. and the 
combined payment $28.40, which is enough to meet 
accrued interest. The amount of 130 dollars for I yr. 
4 mon. is $140.40; $140.40 less $28.40 is L12 dol- 
lars, the third principal. 

The amount of 112 dollars for the last interval, I yr. 
$3 mon. 15 da. is $120.68, the balance due at date of 
settlement. 


2. A note for 120) dollars is dated October 15, 1889, 
Indorsed: October 15, 1890, 1000 dollars; April 15, 1891, 
200 dollars. How much remained due October 15, 1891. 
interest at 6 per cent? $82.56. 

3. A note for 600 dollars is dated July 14, 1890. In- 
dorsed: May 26, 1891, $131,20; December 20, 1892, 40 
dollars; September 14, 1893, 175 dollars. What was due 
July 14, 1894, interest at 6 per cent? $371.70. 


4. A note for 850 dollars is dated January 1, 1892. 
Indorsed: July 1, 1892, $100.62; December 1, 1892, $15.28; 
August 13, 1893, $175.75. What was due on taking up 
the note January 1, 1894, interest at 6 per cent? 

| $650.39. 


5. A note for 400 dollars is dated March 4, 1888. In- 
dorsed: September 4, 1888, 10 dollars; January 4, 1889, 
30 dollars; July 4, 1889, 11 dollars; September 4, 1889, 
80 dollars. What was due March 4, 1890, interest at 6 
per cent? $313.33. 


INTEREST. 5Y 


6. A note for 1750 dollars is dated November 23, 
1892. Indorsed: November 26, 1894, 500 dollars; July 
19, 1895, 50 dollars; September 2, 1895, 600 dollars, De- 
cember 29, 1895, 75 dollars. What was due February 
11, 1896, interest at 7 per cent? $879.71. 

7. A note for 2150 dollars is dated September 20, 1893. 
Indorsed: December 15, 1893, 75 dollars; February 4, 
1894, 200 dollars; April 3, 1894, 150 dollars; July 1, 1894, 
500 dollars: December 16, 1894, 1000 dollars. What was 
due March 20, 1895, interest at 8 per cent? $439.28, 


DISCOUNT. 


>< 


168. Discount is a deduction made from a sum of 
money to be paid. 


169. Commercial Discount is a deduction from 
the price of an article or from a _ bill without regard to 
time. 

170. The Net Price is the list price less the dis- 
count. 


17. 1. What is the net price of a parlor organ, 
listed at 300 dollars, with 50, 20, and 10 off? $108. 
PROCESS. 


00 x .80 x .90 = .360600 x 300 = $108. 


ExpLanation:—When several discounts are allowed 
from list prices, subtract each from 100 and multiply 
the remainders and the list price together to find the 
selling price. . | 


2. Find the value of a bill of goods amounting to 


275 dollars at 5 per cent discount? $261.25. 
3. What is the net price of a bill of hardware, listed 
at 80 dollars, with 25, 20, and 10 off? $43.20 


4, If shoes marked $2.50 per pair are sold at 10 per 
cent discount, what is the net price? $2.50. 


DISCOUNT. 61 


7. Sold county warrants amounting to 30 dollars at 

5 per cent discount. What were the proceeds? 
$28.50. 

10. A merchant receives 2% off for cash. How much 
does he save by discounting a bill of 240 dollars? 

$4.80. 

5. What is the cash value of a bill of goods amount- 
ing to 500 dollars with 20 per cent discount, and 5 off for 
cash? $380. 

8. Ifamerchant sell goods at the catalogue price 
from which he receives a discount of 20 per cent, what 
per cent does he make? 25. 

6. A merchant buys goods at a discount of 20 per 
cent from list price and sells them at 20 per cent above the 
list price. What per cent does he make? 50. 


BANK DISCOUNT: 


172. Bank Discount is simple interest, paid in 
advance, for three days more than the specified time. 


173. The Proceeds of a note are the face of the 
note, less the discount. 

i174. The Maturity of a note occurs on the last 
day of grace, but if the last day of grace fall on Sunday 
or a legal holiday, the note matures on the preceding 
day. 

1. What is the bank discount on $74.16 for 3 months, 


18 days, at 5 per cent? $1.14. 
2. What is the bank discount on 3860 dollars for 5 
months, 8 days, at 6 per cent? $9.66. 


8. Whatis the bank discount on 48 dollars for 4 
months, 12 days, at 8 per cent? $1.44. 


62 DISCOUNT. 


ERU EAD TSS@UNGE 


175. True Discount is the difference between 
the face of a debt and its present worth. 


176. The Present Worth of a debt, due at some 
future time without interest, is the sum which put at 
interest at the specified rate, will amount to the debt 
when it becomes due. 


1L7¢@. To Find the present worth, divide the given 
sum by the amount of one dollar for the given time and 
rate. 


1. When money is worth 6 per cent, what is the pres- 
ent worth of 1300 dollars, payable in 5 years? $1000. 
2. Bought 2895 dollars worth of goods, on two year’s 
credit. What sum will pay the debt now, if money is 
worth 5 per cent? $2631.82. 
3. Bought a house for $975.50, payable in 18 months 
without interest. How much will I gain by paying the 
debt now, money being worth,6 per cent? $80.55. 
4. How much will I gainif instead of paying 5400 
dollars cash for a piece of property, I pay 6000 dollars 
in 16 months, money being worth 9 per cent. 42.86. 


PROPORTION. 


17s. Proportion is an equality of ratios. 


179. Ratio is the relation of one number to another 
of the same kind. 


180. The Extremes of a proportion are the first 
and fourth terms. 


181i. The Means of a proportion are the second 
and third terms. 


182. The Sign of Proportion is the double colon, 
::, or the sign of equality, =. 


183. Simple Proportion is employed for the 
solution of problems in which three quantities are given, 
so related that a fourth may be determined from them. 


184. 1. If six menearn 75 dollars in a week, how 
much will 10 men earn in the same time? $125. 


PROCESS. 

mh) 

6 10 
ExpbLanation:—-Write the third term, or that number 
which is the same kind as the answer, on the right. 
When the result is to be larger than the third term, 
place the larger of the other two numbers on the right 
and the smaller on the left, but reverse this order when 


64 PROPORTION. 


the result is to be less than the third term. Apply can- 
cellation and the result will be the required term. 


2. If 12 yards of cloth cost 15 dollars, what will be 


the cost of 16 yards? $20. 
3. if an ocean steamer sail 1820 miles in 5 days, how 
miles will she sail in 64 days? 2366. 


4. If90 bushels of oats supply 40 horses 6 days, 
how many days will 450 bushels supply them? 30. 

5. If 28 men mow a field of grain in 12 days, how 
many men will be required to mow it in 8 days? = 42. 

6. Ifasum of money at interest produce 12 dollars 
in 4 years, how much will it produce in 7 years? $21. 

7. If 24 bushels of wheat can be bought for 27 dol- 

lars, how many bushels can be bought for 45 dollars? 

40, 

8. If 20 bushels of wheat make 5 barrels of flour, 

how many bushels will it require to make 16 barrels? 

64, 

9. If65 bushels of potatoes can be raised on 24 acres 

of ground, how many bushels can be raised on 7 acres? 
182. 

10. If 35 head of cattle eat 36 acres of grass in a 

- month, how many cattle would 468 acres keep the same 
time. 455, 

11. If 6 men can ean do a piece of work in 45 days, 

how many days will it take 15 men to do the same work? 
18. 

12. If it require 12 men to lay a certain number of 

bricks in 16 days, how many days will it take 8 men to 
lay the same number? 24. 

13. The shadow of a certain tree measures 100 feet, 

while the shadow of a 4-foot stick measures 5 feet. 
What is the height of the tree? 80 ft. 

14. If 18 bushels of wheat is bought for 24 dollars, 

and sold for 80 dollars, how much will be gained on 57 
bushels, at the same rate of profit? $19. 


-PROPORTION. 65 


15. If 20 men can perform a piece of work in 15 days, 
how many men must be added that the work may be 


performed in 4 of the time? 5, 
16. If 4 of a bushel of peaches cost $0.52, what part of 
a bushel can be bought for $0.3? ts. 


17. If butteris worth 18 cents per pound, and 36 
pounds of sugar is exchanged for 30 pounds of butter, 
what is the price of the sugar per pound? $O.15. 


COMPOUND PROPORTION. 


185. Compound Proportion is employed in 
the solution of problems in which the required term de- 
pends on a compound ratio. 


186. In compound proportion, all the terms are in 
couplets or pairs of the same kind, except one. This is 
called the Odd Term, and is always the same kind as 
the answer. Each couplet should be considered sepa- 
rately in making the statement. 


87. 1. If8men earn 40 dollars in 3 days, how 


much will 9 men earn in 4 days? $60. 
PROCESS. 
40 
8/9 
; 3 | 4 


ExpLanation:—Use the vertical form of cancellation, 
writing the odd term on the right; then take the other 
numbers in pairs, or couplets of the same kind, and _ar- 
range them as in simple proportion. 


2. If36 men earn 324 dollars in 18 days, how much 
will 42 men earn in27 days? $567. 

3. If 12 horses plow 11 acres in 5 days, how many 
horses will plow 33 acres in 18 days? 10. 


66 PROPORTION. 


4. If6 men, in 10 days build a wall 20 feet long, 3 
feet high, and 2 feet thick,in how many days can 15 
men build a wall 80 feet long, 2 feet high, and 3 feet 
thick? 16. 

5. Ifaman travel 130 miles in 3 days, when the days 
are 15 hours long, how many days will it take him to 
travel 390 miles, when the days are 9 hours long? 15. 

6. If 12 men can mow 80 acres of grass in 6 days, 
how many days will it take 15 men to mow 200 acres? 

12. 

7. If6men can dig a trench 20 rods long, 6 feet 
deep, and 4 feet wide, in 16 days, working 9 hours per 
day, how many days will it take 24 men to dig a trench 
200 rods long, 8 feet deep, and 6 feet wide, working 8 
hours per day? 90, 


COMPOUND NUMBERS. 


a 


188. Special rules are unnecessary for operations in 
Compound Numbers as the method is the same as 
the corresponding process in simple numbers, the only 
difference, being in their scales of increase. The student 
will readily understand the method of reduction upon 
examining the tables. 


AVOIRDUPOIS WEIGHT. 


189. Avoirdupois Weight is used in weighing 
all coarse and heavy articles, as hay, grain, groceries, etc., 
and all metals except gold and silver. 

190. The Avoirdupois Pound contains 7000 
grains Troy, while the Troy pound contains 5760 grains. 


TABLE. 
T. LB. OZ. 
1 2000 =. 32000 
1 16 
1, What cost 5 pounds of indigo at 10 cents per 
ounce? $8. 


2. What cost 25 lb. 8 oz. of butter at 16 cents per 
pound? $4.08. 


68 COMPOUND NUMBERS. 


3. What cost 4500 pounds of hay at 6 dollars per 


ton? $13.50. 
4, A horse weighs 1440 pounds Avoirdupois. How 
much would he weigh by Troy weight? 1750. 
TIME, 


191. The Table given below is sufficiently accurate 
for ordinary business purposes. 


TABLE. 
YR. MON. DA. HR. 
1 12 360 8760 
1 30 720 
1 24 
1. I was born April 25, 1869. How old was I, July 
14, 1890? 21 yr. 2 mon. 19 da. 


2. The Declaration of Independence was written 
July 4, 1776. How many years had elapsed, March 1, 
1860? 83 yr. 7 mon. 27 da. 


LONG MEASURE. 


192. Long Measure is used in measuring lenghts 
and distances. 


TABLE. 
MI. RD. YD. FT. IN. 
if 320 1776 5280 63360 
1 at 163 198 
1 3 36 
1 12 
1. Reduce 264 ft. to rods. 16. 


2. Reduce 2 mi. 2 rd. 2 ft. to feet. 10595, 


COMPOUND NUMBERS. 69 
3. How many steps of 2 ft. 8 in. each will a man take 


in walking 2 miles? 3960. 


SURVEYOR S MEASURE. 


193. Surveyor’s Weasure is used in measuring 
land, laying out roads, establishing boundaries, ete. 

194. Surveyors use the Gunter’s Chain, which is 
4 rods long and contains 100 links. 


TABLE. 
MI. CH. RD. LK. IN, 
] 80 320 8000 63360 
1 4 100 792 
1 25 198 
1 7.92 
1. Reduce 400 links to rods. 16. 
2. Reduce 128 rods to chains. 32. 
3. Reduce 640 chains to miles. 8. 
4. Reduce 3 ch. 25 links to rods. 1% 


SQUARE MEASURE 


195. Square Measure is used in measuring sur- 
faces. 


TABLE. 
A. SQ. RD. SQ. YD. * --->SQ. FUE SQ. IN. 
1 160 4840 43560 6272640 
] 305° 2724 39204 
1 9 1296 


1 144 


70 COMPOUND NUMBERS. 


1. Reduce 480 square rods to acres. 3. 
2. Reduce 2560 square rods to acres. 16. 
3. Express ~ of an acre in square rods. 60. 
4. How many square feet in 27 square yards? 243. 

». How many square feet in 1728 square inches? 
12. 

6. How many square yards in 4545 square feet? 
505. 


7. A gentleman divided his farm of 328 A. 74 sq. rd. 
equally among his 3 sons. What was the share of each? 
109 A. 78 sq. rd. 


CUBIC MEASURE. 


196. Cubic Measure is used in measuring 


solids. 
TABLE. 
CU. YD. CU. FT. CU. IN. 
1 27 46656 
1 1728 
1. Reduce 20736 cubic inches to cubic feet. 12. 
2. Reduce 482 cubic feet to cubic yards. 16. 


3. Reduce 2 cu. yd. 2 cu. ft. to cubie inches. 
. 96768. 


DRY MEASURE. 


197. Dry Measure is used for measuring grains, 
vegetables, fruits, ete. 

198. The Standard Unit of dry measure is the 
bushel which contain 21502 cubic inches, or nearly 14 
cubic feet. 


COMPOUND NUMBERS. 71 

199. The Standard Gallon of the United States 

contains 231 cubie inches, or about -"s of a ecubie foot. 
The dry gallon contains 268% cubic inches. 


TABLE. 
BU. PK, GAL. QT. PT 
1 4 8 3p 64 
1 2 8 16 
1 4 8 
1 2 
1. Reduce 448 pints to bushels. (ep 


2. Reduce 1 bu. 1 pk. 1 gal. 1 qt. 1 pt. to pints. 
91. 


200. The weight of a Bushel of various articles 
is given in the following 


TABLE. 

ARTICLES. ILB ARTICLES, LB. 

| Apples 50 Hair, unwashed | 8 
Barley 48) Hemp seed 44 

| Beans 60 Hungarian seed 45 
| Bluegrass seed 14) Lime 80 
Bran 20, Millet 45 
Buckwheat 52) Oats 132 
Castor beans 46) Onions 157 
Charcoal 22|| Onion sets ~~ =—«(14 
Clover seed 60}, Potatoes 60 
Coal 80), Potatoes, sweet 50 

| Corn 56|| Rye 56 
Corn, in ear 70), Salt 50 
Corn meal 50) Timothy seed 45 

| Flax seed 56 Turnips 56 
| Hair, washed 4|| Wheat 60 


1. How many bushels of corn in a load weighing 
1344 pounds? 24. 

2. How many bushelsin a load of wheat weighing 
1290 pounds? 214. 


Lo COMPOUND NUMBERS. 


3. How much’should I receive for 1536 pounds of 
oats at 20 cents per bushel? ~ $9.60. 
4. Bought 12 gallons of syrup at 75 cents a gallon, 
and sold it at 24 cents a quart. How much did I gain? 
$2.52. 
5. At $2.40 per bushel what should I receive for a 
load of timothy seed weighing 2271 pounds, deducting 
the weight of the wagon, 1236 pounds? $55.20. 


MISCELLANEOUS TABLE. 


4 Inches, 


1 Shingle. 


4 Inches, 1 Hand. 
6 Feet, Ll Fathom. 
3 Miles, 1 Weague. 
4 Gills, LEP ing 
5 Bushels corn, 1 Barrel. 
60 Seconds, 1 Minute. 
60 Minutes, 1 Hour. 
16 Drams, 1 Ounce. 
24 Sheets, 1 Quire. 
20 Quires, 1 Ream. 
12 Things, 1 Dozen. 
12 Dozen, 1 Gross. 
20 Things, 1 Score. 
15° Longitude, 1 Hour. 
100 lb. Grain, 1 Cental. 
100 lb. Fish, 1 Quintal. 
196 lb, Flour, 1 Barrel. 
200 Ib. Beef or pork, 1 Barrel. 
280 lb. Salt, 1 Barrel. 
39.37 Inches, 1 Meter. 


4.8665 Dollars, 1 £ Sterling. 


INVOLUTION. 


202. Involution is the process of raising a num- 
ber to any given power. 

203. A Power is the product arising from multi- 
plying a number by itself in continued multiplication. 

204. The First Power of a number is the number 
itself. 


205. The Second Power of a number is called its 
SQUARE. 

206. The Third Power of a number is called its 
CUBE. 

207. The Degree of a power is indicated by an 
exponent, which is a small figure placed a little above 
and at the right of the number. Thus, 3*, indicates 
the fourth power of 3. 

208. The Product of any two or more powers is 
the power denoted by the sum of their exponents. Hence 
if we multiply the third power of a number by the fourth 
power the product will be the seventh. 


209. 1. What is the fourth power of 3? 81. 
PROCESS. 
$3x3x3xS=SL. 


ExpLanation:—Multiply the number successively by 


74 INVOLUTION. 


itself till it has been taken as many times as a factor as 
there are units in the exponent of the required power. 


2, What is the square of 16? 256. 
3. What is the cube of 9? 729. 
4. What is the fourth power of 5? 625. 
5. What is the fifth power of 6? RiA0; 


NUMBERS ENDING WITH «3. 


210. The following method of squaring a number 
whose Unit Figure is 5 will be found valuable. 


211° 1. What is the square of 25? 625. 
PROCESS. 


25 
20 


625 


Expianation:—Multiply the part preceding 53 by 
itself increased by I and prefix the result to 25. 


2. What is the square of 75? 5625, 
3. What is the square of 35? 1225. 
4. What is the square of 45? 2025. 
>. What is the square of 195? 38025. 
6. What is the square of 115? 13225. 
7. What is the square of 995? 990025. 


ENTE GWE 2's: 


212. The Square of numbers ending with 25 may 


—~l 
A 


INVOLUTION. 


be readily written out by the method explained below. 


213. 1. What is the square of 625? 390625, 
PROCESS. 
39 0625 

ExpLaNnaTion:—Square the part preceding 25, add 


half the same part to the result, discarding fractions, 
and annex 0625, or if the part preceding 25 is odd, 
annex 3625. 


2. What is the square of-425? 180625. 
3. What is the square of 925? 855625. 
4. What is the square of 325? 105625. 
5. What is the square of 1025? 1050625. 


MIXED NUMBERS. 


214. The method of squaring Mixed Numbers 
ending with 4 is illustrated below. 


215. 1. Find the square of 63. 42 


vol 


PROCESS. 
6 
62 


A424 

ExpLaNnaTion:—We say @ times 6 are 42 and annex 
4 to the product. Always add 1 tothe multiplier. This 
method is applicable in all problems like the above. The 
student should study the method and apply the same 
principle to other fractions. 

2. Find the square of 73. d64. 

3. Find the square of 93 905. 

4. Find the square of 113. 1324, 


76 INVOLUTION. 


5. Find the square of 493. 24504. 
6. What cost 123 pounds of butter at 123 cents per 
pound? $1564. 


SOU ARE-OF PW @SDIGI Ts SE ie. 


216. Small Numbers may be squared mentally, 
by the following simple method. 


217. 1 What is the square of 18? 324. 


PROCESS. 
16 x 20+4= 324 


Expianation:—Take the product of two numbers, one 
of which is as much less than the number to be 
squared as the other is greater, and one of the numbers 
a multiple of ten, and add the square of the difference 
between the givén number and one of the asumed num- 
bers. 


2. What is the square of 27? 729, 
3. What is the square of 21? | 441, 
4, What is the square of 33? 1089. 
». What is the square of 79? 6241. 


DOU ATs DOLE NT IN Bis 


218. The Square of any number of nInES may be 
written out without multiplying. 


219. 1. What is the square of 999? 998001. 
PROCESS. 
998001 


EXPLANATION :—Erase one 9 from the left of the num- 


> ae 
bi Ree 
i i 2 ae ed 8 
a) 


A ~ < 


“ber to be Re a wtinted annex an $8, as many ciphers as there 
e nines, anda I. ; 
2. Find the square of 9. 81. 

3. Find the square of 99. 9801. 
4. Find the Square of 9999. 99980001. 


" INVOLUTION. | 77 ie 
y 
: 


poe 


aa se 


EVOLUTION. 


220. Evolution is the process of finding roots of 
numbers. 

221. A Root of a number is one of its equal fac- 
tors. 

222. The Sign of Evolution, \, is a modification 
of the script letter r. Roots are also indicated by frac- 
tional exponents. 

223. The following table of Squares and Cubes 
should be learned by the pupil. 


A Maylide 


Numbers 1, 2, 3, 4 5, 6, 4) othe 
Squares I, 4, 9, 16, 25, 36, 49, 64, SI. 
subes 1, 8, 27, 64, 125, 216, 343, 512,729. 


224. It will be observed that Square Numbers 
never end in 2, 3, 7, or 8, and that the cubes of no two 
digits end with the same figure. Hence, in finding the 
eube root of perfect cubes, we can easily determine the 
unit figure of the root from the unit figure of the power. 


———— cle en SS aS ns eS OTS 


Ee eee 


EVOLUTION. 79 


EVOLUTION BY FACTORING. 


225. To find Any Root of a perfect power, resolve 
the number into its prime factors, and for the square 
roct take one of two equal factors, for the cube root * 
take one of three equal factors, ete. 


226. 1. What is the square root of 225? 15. 
PROCESS. 
225 =—-383x3xkK3x5 


225 = $8 X53 = 15 


2. Find the square root of 625. 25. 
3. Find the square root of 1296. 36. 
4. Find the cube root of 1728. 12. 
5. Find the fourth root of 1296. 6. 
6. Find the fifth root of 243. 3 


SOUARE ROOT BY ANALYSIS. 


227. Separate into periods of two figures each, 
beginning at the right. The first figure is the root of 
the greatest square in the left hand period. Subtract 
its square from the first period, and bring down the 
next period. Double the root found and divide, disre- 
garding the right hand figure of the dividend, and place 
the result in the root and at the right of the divisor. 
Multiply the complete divisor by the last figure of the 
root and proceed as before. If a cipher occur in the 


. root, annex a cipher also to the trial divisor and bring 


down the next period. 


80 EVOLUTION. 
228. 1. What is the square root of 1296? 36. 
PROCESS. 


12.96(36 


66 396 


396 
2. What is the square root of 2304. 48, 
3. The area of asquare field is 6561 square rods. 
How many rods in length or breadth? 81. 
4, A man has a square field containing 4096 square 
rods. How many rods in length or breadth? 64. 


5. The length of a rectangular field containing 20° 
acres is twice its width? What is the distance around 
it? 240 rd. 

6. A square field measures 6 rods on each side. What 
is the length of the side of a square field which is 16 
times as large? 24 rd. 

7. If it costs $572 to enclose a field 72 rods long and 
32 rods wide, how much less will it cost to enclose a 
square farm of equal area with the same kind of fence? 


SIMILAR SURFACES. 


229. Similar Surfaces are to each other as the 
squares of their like dimensions. 


230. Like Dimensions of similar surfaces are 
to each other as the square roots of their areas. 


231. 1. A hole made by a 2 inch auger bit, is how 
many times as large as one made by an inch auger bit? 
4, 


EVOLUTION. 81 


Cal 


2. If the area of a circle, whose diameter is 7 feet is 
38.5 sq. ft., what will be the area of a circle 21 feet in 
diameter. 346.5 sq. ft. 

3. A rectangular field is 12 rods wide and 20 rods 
long. What must be the width of a road across one end 
and one side to contain 4 the area of the entire field? 

2 rd. 

4. If one side of a triangle is 12 feet, and its area is 
36 square feet, how many square feet in the area of a 
similar triangle, the corresponding side of which is 8 
feet? 16. 

5. The area of a triangle, the length of whose base is 8 
rods, is 92 square rods. How many square rods are there 
in the area of a similar triangle, the corresponding side 
of which is 4 rods? 13. 


Cay Bs ROOTBY INSPECTION: 


232. The Cube Root of perfect cubes of not more 
than six figures can be easily found by inspection. 


233. 1. Find the cube root of 15625. 25. 
PROCESS. 
( 15,625 )* = 25 


ExpLaNnation :—-For the first figure of the root, we write 
the root of the greatest cube in the left hand _ period, 
which is 2, and then by inspection, or reference to the 
table, we see at once that the other figure of the root must 
be 5, as the cube of no other digit ends with 5. 


2. Find the cube root of 1728. 12. 
3. Find the cube root of 12167. 23. 
4. Find the cube root of 39304. 34. 
5. Find the cube root of 91125. 45. 


82 EVOLUTION. 
5. Find the cube root of 175616. 56. 
6. Find the cube root of 300763. 67. 
7. Find the cube root of 474552. 78. 
8. Find the cube root of 704969. 89. 
9, Find the cube root of 753571. 91. 


CUBE; RG) Ochs bY aac eas less 


234. Separate into periods of three figures each. 
The first figure is the root of the greatest cube in the left 
hand period. Subtract the cube and bring down the 
next period. Square the root found, multiply by 300, 
and divide to find the second figure of the root. To three 
times the first figure of the root, annex the last. Multi- 
ply this factor by the last root figure and add the result 
to the trial divisor. Multiply the complete divisor by 
the last figure of the root, subtract and proceed as_be- 
fore. When the dividend will not contain the trial divi- 
sor, write a cipher in the root and two at the right of the 
trial divisor. 

235. 1. What is the cube root of 13824? 24. 

PROCESS. ) 


13,824(24 
‘s 


1200 3824 
256 F 
1456 5824 
2. What is the cube root of 74088? 42. 


8. What is the side of a cubical box which contains 
873248 solid inches? 6 ft. 


EVOLUTION. 83 


4. What is the depth of a cubical bin whose contents 


are 79507 cubic feet? 43 ft. 
5. What is the length of the side of a cubical box that 

contains 15625 cubic feet? 25 ft. 
6. What is the side of a cube equal to a pile of wood 

81 feet long, 27 feet wide and 9 feet high? 27 it. 


SIMILAR SOLIDS. 


236. The Contents of similar solids are to each 
other as the cubes of their like dimensions. 


237. Like Dimensions of similar solids are to 
each other as the cube roots of their contents. 


238. The Side of a cube, whose solidity bears a giv- 
en relation to that of a cube whose side is given, is found 
by eubing the given side, multiplying the result by the 
given proportion and extracting the cube root of the 
product. 


239. 1. What is the side of a cubical vat which 
contains } as much as one whose side is 6 feet? 3 ft. 

2. Ifa cubic inch of gold is worth 200 dollars, what 
is the worth of a cube of gold whose side is 3 inches? 
) $5400. 


3. Ifa cubical block of granite, whose side is 4 inches 
weigh 12 pounds, what will a cubic foot of the same gran- 
ite weigh? 324 Ib. 

4, I have a cubical box whose side is 3 feet. I want 
another which will contain 8 times as much. What will 
be the length of its side? 6 ft. 

5. Ifacannon ball 8 inches in diameter weigh 40 
pounds, what is the weight of one of the same metal, 
whose diameter is 4 inches? 5 Ib. 


MENSURATION. 


240. For the practical convenience of those who have 
occasion to refer to Mensuration, we give the follow- 
ing principles, covering the whole ground of practical 
geometry. 


PRINCIPLES. 


241. The Diagonal of a square is equal tothe side 
of the square multiplied by 1.414. 


242. The area of a Triangle is equal to half the 
product of the base by the altitude. 


2438. The side of an Inscribed Square is equal 
to the diameter multiplied by .7071. 


244. The area of any Parllelogram is San to 
the product of the base by the altitude. 


245. The areaof a Parabola is equal to the base 
multiplied by two-thirds of the altitude. 


246. The area of an Ellipse is equal to the product 
of the two diameters multiplied by .7854. 


247. The contents of a Sphere is equal to the 
cube of the diameter multiplied by .5236. 


MENSURATION. 85 


248. The contents of a Wedge is equal to the area 
of the base multiplied by half the altitude. 


249. The surface of a Sphere is equal to the 
square of the diameter multiplied by 3.1416. 


250. The area of a Sector of a circle is equal to the 
length of the are multiplied by half the radius. 


251. The area of a Trapezoid is equal to its alti- 
tude multiplied by half the sum of its parallel sides. 


252. The side of an Imnseribed Equilateral 
Triangle is equal tothe diameter multiplied by .866025, 


253. The contents of a Cylinder or Prism is 
equal to the area of the base multiplied by the altitude. 


254. The contents of' a Pyramid or Cone is 
equal to the area of the base multiplied by one-third of 
the altitude. 


255. The convex surface of a Pyramid or Cone 
is equal to the perimeter of the base multiplied by half 
the slant height. 


256. The entire surtace of a Cylinder or Prism 
is equal to the area of both ends plus the product of the 
length by the periphery. 


257. The side of a Cube which may be cut from a 
given sphere is equal to the square root of one-third of 
the square of the diameter. 


258. The diameter of a Cirele that shall contain 
the area of a given square is equal to the side of the 
square multiplied by 1.1284. 


259. The area of any Regular Polygon is equal 
to the perimeter multiplied by half the perpendicular 
distance from the center to one of the sides, 


86 MENSURATION. 


260. The convex surface of a Frustrum of a pyra- 
mid or cone is equal to the sum of the perimeter of the 
two bases multiplied by half the slant height. 


261. The area of a Trapeziumi is equal to the di- 
agonal multiplied by half the sum of the perpendiculars 
drawn from the vertices of the opposite angles to the 
diagonal. 


262. The Ratio betewen the diameter and circum- 
ference of a circle, expressed decimally and the approxi- 
mation carried to thirty places. is 3.14159265358979323846 
264338328, 


263. The area of a Segment ofa circle is equal to 
the area of a corresponding sector less the area of the tri- 
angie, or plus the area of the triangle when the segment 
is greater than a semicircle. 


264. The side ofa Square that will contain the 
area of a given circle is equal to the square root of the 
area, or the diameter multiplied by .8862, or the circum- 
ference multiplied by .2821. 


265. The contents of a Frustrum of a pyramid 
or cone is equal to the sum of the areas of the two ends 
plus the square root of the product of these areas, multi- 
plied by one-third of the altitude. 


RECTANGISES. 


266. A Rectangle is a figure that has four straight 
sides and four right angles. 


267. The Area of a rectangle is equal to the pro- 
duct of its length by its breadth. 


MENSURATION. 87 


268. 1 At 12 cents a square yard, what will it cost 
to paint the walls of two rooms, each 16 feet square and 


9 feet high? ~ $15.36. 
STATEMENT. 
| 128 
9/9 
12 


ExpLanation:—-Multiply the entire distance around 
the rooms by the height, divide by 9 to reduce to square 
yards, and multiply by the price per yard. 


2. How many square feet in a floor 16 feet long and 


14 feet wide? 224. 
3. How many acres in a field of land 96 rods long 
and 80 rods wide. 48, 
4. Ifa floor is 12 feet long, how wide must it be to 
contain 132 square feet? 11 ft. 
5. Find the difference between a floor 20 feet square, 
and two others 10 feet square. 200 sq. ft. 
6. One side of a rectangular field containing 63 acres 
is 120 rods. What is the other? 84 rd. 
7. How many yards of carpet 13 yards wide will cover 
a floor 18 feet long, 15 feet wide? 20 yd. 


8. At 10centsa square yard, what will it cost to 
plaster the walls and ceilings of three rooms, each 15 ft. 
6 in. long, 13 ft. 8 in. wide and 12 ft. high? $44.51. 


PE tAN GPS, 


269. A Triangle is a figure which has three an- 
gles and three sides. 


270. A Right Angle is the angle formed when 
one line is drawn perpendicular to another. 


271. The Hypotenuse of aright angled triangle is 
the side opposite the right angle. 


88 MENSURATION.. 


272. The Base of a triangle is the side on which it 
is assumed to stand. 


273. The Perpendicular is the side which forms 
a right angle with the base. 


274. The Area ofa triangle is equal to half the 
product of the base by the altitude. 


275. The Hypotenuse of aright angled triangle, 
is equal the square root of the sum of the squares of the 
other two sides. 


276. The Base or Perpendicular is equal to 
the square root of the difference of the squares of the 
hypotenuse and the other side. It is also equal to the 
square root of the product of the sum and difference 
of the hypotenuse and the other side. 


2@¢@. 1. What is the area of a triangle whose base 
is 16 feet and whose altitude is 13 feet? 96 sq. ft. 
2. ‘The base of a right angled triangle is 40 feet, and 
the perpendicular is 30 feet. What is the eb aor 
5 
3. The hypotenuse of a right angled triangle is 73 
feet and the perpendicular is 43 feet. What is the base? 
6 ft. 
4, The base of a right angled triangle is 34 feet and 
the hypotenuse is 123 feet. What is the perpendicular? 
ies 
5. A rectangular field is 60 rods long and 45 rods 
wide. What is the distance between two opposite corn- 
ers? 75 rd. 
6. What is the area of a triangular piece of ground 
whose base is 40 rods, and whose perpendicular height 
is 28 rods? 3d A. 
7. A pole is 27 feet high. How many feet above the 
ground must it be broken in order that the upper part, 
clinging to the stump, may touch the ground 9 feet from 
the base? 12. 


MENSURATION. 89 


8. The main mast of a vessel .is 72 feet high. How 
many feet above deck must it be broken in order that the 
upper part, clinging to the stump, may touch the deck 
16 feet from the base? 342, 


CLERC lakes. 


278. A Cirele is a plane figure bounded by a 
curved line, every part of which is equally distant from 
a point within, called the center. 

279. The Circumferenee is the line which bounds 
the circle. 

280. The Radius of a circle is a straight line 
drawn from the center to the circumference. 


281. The Diameter ofa circle is a straight line 
drawn through the center, and terminated by the cir- 
cumference. 

282. The Circumference of a circle is equal to 
the diameter multiplied by 3+. Hence, the diameter is 
equal to the cireumference divided by 3+. 


283. The Area ofa circle is equal to the cireumfer- 
ence multiplied by one-fourth the diameter; or, the 
square of the diameter multiplied by +44. 


284. 1. Whatisthe circumference of a cirele 28 


inches in diameter? 88 in. 
STATEMENT. 
| 28 
7\|22 


EXPLANATION: Simply multiply by 3+, by reducing it 
to an improper fraction and applying cancellation. 

2. What is the circumference of a log 14 inches in 
diameter? 3 ft. 8 in. 

3. How far is it around a circular pond that is 45 feet 
in diameter? 1413 ft. 


90 MFNSURATION. 


4. What is the area of a circle 14 feet in diameter? 


154 sq. ft. 
5. What is the radius of a circle whose circumference 
is 616 feet? 98 ft. 
6. What is the area of a circular pond 70 rods in 
circumference? 3850 sq. rd. 
7. What is the diameter of a circle whose circumfer- 
ence is 154 feet? 49 ft. 


LUMBER MEASURE. 


285. To measure Lumber, multiply the width in 
inches, the thickness in inches, and the length in feet 
together, and divide the product by 12. 


286. To find the quantity of lumber in a Leg, mul- 
tiply the square of the diameter in inches at the small 
end by the length in feet, and divide the product by 24. 


28¢. Whena board Tapers uniformly in width, 
find the average by taking half the sum of the two ends. 
If it taper also in thickness, the contents in board feet 
may be found by multiplying the sum of the areas of the 
two ends in square inches by the length in feet, and 
dividing the product by 24. 


FORMULA. 


No. Pieces. 


Te OO 
atte % 
Mc 


Length. 
| Width. 


Thickness. 


MENSURATION. 9] 


288. 1. How many feet in 3 pieces 2 x 4, 16 feet 


long? 32. 
STATEMENT. | 
33 
2 
4 4 
—«G 


ExpLanation:—Write the given dimensions on the 
right and the factors of 12 on the left. The price per 
foot should also be written on the right when the cost 
is required. 


2. How many feet in 11 pieces 6 X 8, 14 feet long? 


616. 
3. How many feet in 16 boards 10 inches wide and 12 
feet long? 160. 
4. How many feet of lumber in a log 15 inches in 
diameter and 16 feet long? 150. 
5. How many board feet in a post 3 x 4 at one end, 
4 x 6 at the other, and 10 feet long? 15. 
6. How many feet in 56 fence posts 2 x 4 at one end, 
and 4 x 4 at the other, and 8 feet long? ° 448, 
7. What will it cost to floor a room 14 by 20 with 
2 dollar flooring, allowing + for matching? $6.72. 


8. <A farmer bought 75 boards of fencing 6 inches 
wide, and 16 feet long. What did they cost at 2 dollars 
per hundred feet? $12. 

9. How many feet of boards will be required to en- 
close two gable ends of a building 16 feet wide, allowing 
that the roof is half pitch? 128. 


LATH MEASURE. 


289. To Compute the number of lath required to 
cover a wall, multiply the number of square yards by 15. 


Q? MENSURATION. 


290. In Computing the contents of walls, plas- 
terers make no allowance for doors and windows. 


* FORMULA. 


{ 
9 Breadth. . 


(15 


291. 1. How many lath will be required for a wall 


containing 250 yards? 3750. 
2. How many lath will it take to cover the ceiling of 
a room 24 feet long and 16 feet wide? 640. 


3. How many lath will it take to cover the walls of a 
room 22 feet wide, 32 feet long, and 10 feet high? 


1800. 
4, At $2.80 per thousand, what cost the lath for the 


walls and ceiling of a room 12 by 15, and 8 feet 4 inches 
high? $2.94. 


SHINGLE MEASURE. 


292. ‘To find the length of the Rafters, giving the 
roof one-third pitch, take + of the width of the building: 
for one-half pitch take } of the width of the building. 


293. It Requires 9 shingles to the square foot if 
exposed 4 inches; 8 if exposed 43 inches; 7 if exposed 5 
inches, and 6 if exposed 53 inches to the weather. 


i i 


MENSURATION. 93 


294. 1. Find the number of shingles required to 
make a roof 16 feet long and the rafters on each side 12 
feet, shingles exposed 43 inches, 3072. 

2. What length of rafters must | buy for a building 
28 feet wide, giving the roof half pitch and allowing 1 
foot for projections? 22 ft. 

3. What length of rafters will be required for a 
building 15 feet wide, giving the roof one-third pitch 
aud allowing 1 foot for projections. 10 ft. 

4. At $2.50 per thousand, what cost the shingles for 
a third pitch roof on a building 20 x 382 allowing 6 in- 
ches all around for projections, shingles exposed 44. in- 
ches? 3 $16.50. 

5. How many shingles will be required for a_ half 
pitch roof on a building 16 x 20, allowing for projections 
| foot on each side and six inches até each end, shingles 
exposed 4 inches? 4914. 


WOOD MEASURE. 


295. A Cord of Wood contains 128 cubic feet, 
To find how many cords in a pile of wood, multiply the 
length, height, and width, together, and divide the 
product by 128. 


FORMULA. 


94 MENSURATION. 


296. 1. How many cords in a pile of wood 32 feet 


long, 8 feet high, and 4 feet wide? 8. 
STATEMENT. 
32 
128 | 8 
4 
2. How many cords in a range of wood 32 feet long, — 
6 feet high and 4 feet wide? 6. 
3. How many cords ina pile of wood 28 feet long, 
16 feet wide, and 12 feet high? 42. 
4. What is the worth of a pile of wood 16 feet long, 
4 feet wide and 5 feet high, at $22 per cord? $6. 
5. Ifa pile of wood is 32 feet long, and 4 feet wide, how 
many feet high must it be to contain 12 cords? 12; 


6. A man sawed a pile of wood 40 feet long, 4 feet wide 
and 53 feet high, for $1.50 per cord. How much did he. 
earn? $10. 


FLAY NER A Us 


297. The only Accurate Method of measuring 
hay is to weigh it; but the method explained below will 
be found sufficiently accurate for ordinary purposes. 


FORMULA. 


Width. 


700) Height. 


Exprianation:—To find the number of tons in ordinary 


MENSURATION. 95 


stacks or ricks, multiply the length, width and entire 
height together and divide the product by 700. 


298. 1. How many tons of hay in a rick 10 feet 


wide, 14 feet high and 20 feet long? 4, 
STATEMENT. 
7| 10 
1d) 14 
10 | 20 
2. How many tons in a stack of hay 12 feet wide, 15 
feet high and 28 feet long? 73 
3. How many tons in a stack of hay which measures 
12 feet wide, 14 feet high and 15 feet long? 33 
MASONRY. 


299. Brickwork is commonly estimated by the 
thousand bricks. 


300. Stone Masonry is estimated by the perch, 
which contains 25 cubic feet. 


301. In Computing the contents of walls, masons 
make no allowance for the corners, but estimate their 
work by the entire distance around on the outside. They, 
however, make a deduction of half the space occupied 
by windows, doors, and other openings. 


302. 1. How many bricks will it take to build a 
wall 40 feet long, 84 feet high, and 1} feet thick, allow- 
ing 20 bricks to the cubic foot, and deducting +) for 


mortar? 9000. 
STATEMENT. 
AO 
3 | 25 
Bié< 
20 


56 MENSURATION. 


2. How many perches of masonry in a wall 35 feet long, 


15 feet high, and 3 feet thick. 63. 
STATEMENT, 
3 
25 | 15 
3 
3. How many perches in a wall 30 feet long, 165 feet 
high, and 4 feet thick? 0, = 


4. At $2.25 a perch, what will it cost to build the 
walls of a cellar 160 feet in circumference, 12) feet high. 
2 feet thick, allowing ;'5 for mortar and filling? $324. 

5. How many bricks will be required to build an en- 
closure 12 feet wide and 18 feet !ong, the walls of which 
are 10 feet high, and 18 inches thick, allowing 20 bricks 
to the cubic foot? 18000. 


BOX MEASURE. 


303. To find the contents of any regular vessel in 
Bushels, multiply the product of its length, breadth, 
and depth by .8; or, by .4 for bushels of corn in the ear; 
or, by 74 for the contents in gallons. If the vessel is 
cylindrical, proceed as if it were square and take 4 of 
the result. 


FORMULA. 
5° 4 
; Length. 
; Breadth. 


{ 
) 
? | 
ickmess. ; 
Thickne pi 


a 
a 


ExpLaNnaTion: —In the above formula, we have the com- 


MENSURATION. 97 


mon fraction $ intead of .8, either being correct. For 
corn in the ear, write 2 on the right instead of 4. 


304. 1. How many bushels ina bin 10 feet long, 


2 feet 6 inches wide, and 1 foot 3 inches deep? 25, 
STATEMENT. 
| 10 
2,5 
4| 5 
a 


ExpLanation:—Reduce inches to the fraction of a foot 
and apply cancellation. Consider .8 as a whole number 
and point off one figure from the right of the result. 


2. How many gallons in a box 10 feet long, 5 feet 


wide, and 4 feet deep? 1500. 
3. How many bushels in a cylindrical tub 5 feet 
in diameter and 6 feet deep? 96. 
4. How many bushels in a wagon box 10 feet long, 
3 feet wide, and 21 inches deep? 42. 
5. How many bushels of corn, in the ear, ina crib 14 
feet long, 10 feet wide, and 8 ft high? 448, 


6. How many bushels in a box 7 feet 6 inches long, 
4 feet 8 inches wide, and 2 feet 6 inches deep? 70. 

7. What will be the cost of the wheat at 75 cents per 
bushel, which would just fill a hogshead 3 feet 6 inches 
in diameter, and 4 feet 8 inches deep? $27.44. 

8. How much would the wheat be worth at $1.25 per 
bushel which would just fill a bin, the base of which is 
123 feet square, and the height 10 feet? $1562.50. 


Our CURIOSITY SHOP. 


a a ee 


305. The Following Paragraphs will be devoted 
to a brief exposition of many interesting and valuable 
properties of numbers, wonders, curiosities, ete. 


LIGHTNING ADDITION. 


306. This is the Method used by most lightning 
calculators throughout the country. 


PROCESS. 
746327 

253673 ¥ 
170624 . 

$29376 

263405 

724291 


= SS. 


2987696 


ExpLaNnaTion:—The first and second lines are so ar- 
ranged as to add to 9 all the way through, except in units 
place, where they add to 10. The same is true of the 
next two lines, and so on down till the desired number of 
couplets are written. The last two lines are made up of 
small figures, so arranged that in adding them there will 
be nothing to carry. By adding the two bottom lines 


MISCELLANEOUS. 99 


and prefixing the number of couplets above, the correct 
sum is obtained. The method may be varied to add the 
top and bottom lines, the two middle lines, ete. 


NINE. 


307. Every number consists of a certain number of 
Nines, plus the sum of its digits. From the foregoing 
axiomic principle, an instructive, as well as a very inter- 
esting and amusing exercise may be deduced. If you 
have a friend present, have him write a number with as 
many digits as he may choose. Instruct him to subtract 
the sum of the digits from the number, cancel one figure, 
add the remaining digits, and tell you the result. By 
deducting the result from the first succeeding multiple 
of nine, the remainder will be the figure canceled. 


PROCESS. 
46532 
20 


Expn.anation:—If 16 be given as the sum of the 
digits in the remainder, exclusive of the one canceled, 
then, you would subtract it from 18, the first multiple of 
nine above. Hence, 2 would be the digit canceled. 
This property of nine affords short and accurate tests 
for ordinary operations in Arithmetic. 


Reo LN ON RIGU RE 


308. To produce the result in One Figure, write 
the nine digits, omitting the 8, in their regular order, as 
the multiplicand; then multiply the figure desired in the 
answer by 9, and use this product as the multiplier. 


100 MISCELLANEOUS. 


309. 1. Multiply 12345679 by some number that 
will produce the answer in fives. | 45, 


PROCESS. 
12345679 


45 


61728395 
49382716 


93953955555 


2. -By what number must 12345679 be multiplied to 
produce the answer in sevens? 63, 


AN OU DOR WU AAI ie rob Wor lee 


310. Leta person write any Unknown Number, 
double it, add a given number, multiply by 5, cut off the 
right hand figure, and subtract the first number. 

31. 1. Take 12 for the starting number and 16 
for the number to be added. 8. 


PROCESS. 


Expianation:—The result will be one-half the number 
added, discarding fractions. After you have the result, 


MISCELLANEOUS. 101 


other operations may be performed at pleasure. With 
persons unacquainted with the principle upon which this 
exercise is based, the successful operation of it will arouse 
a vast amount of wonder and curiosity. 


GREGORIAN CALENDAR. 


312. To ascertain the Day of the Week for any 
given date, divide the number of years elapsed since the 
beginning of the century by 4, discarding fractions; to 
this quotient add the dividend, the day of the month, 
the complement of the month, and the complement of 
the century; divide this amount by 7 and the remain- 
der will indicate the day of the week. 


313. The Complement of the present century is 
0, and for 1900 it will be 5. The complements of January 
and February, in leap years, are 2 and 5, respectively. 
The other complements are exhibited in the following 


AeA ie Br 
Jan. 3. Apr. 2. Jul. 2. Oct. 3. 
Feb. 6. May A. Aug. 5. Nov. 6. 
Mar. 6. Jun. 0. Sep. 1. Dee. 1. 
314. 1. On what day of the week did October 26, 
1870, fall? Wednesday. 
PROCESS. 
70 Year 
17 Leap Years 


26 Day of Month 
3 Comp. of October 
0 Comp. of Century 
7)116 Sum 
16-4 Remainder 


: 


102 MISCELLANEOUS. 


EXPLANATION:—The remainder indicates the fourth 
day of the week, or Wednesday. 


2. On what day of the week did May 21, 1836 fall? 


Saturday. 
3. A gentleman was born April 25, 1869. On what 
day of the week did his birthday fall? Sunday. 


MARKING KEYS. 


315. Merchants employ various combinations to 
render cost and price marks unintelligible to all, except 
their employes. Sometimes signs or symbols are used, 
but most business men make use of words or phrases of 
ten letters, each representing a figure. Thus take the 
word 

BRICK HOU SE 


1238 45 6 7 8 9 O 


EXPLANATION :—Suppose we wish to mark a piece of 
goods at, say 15 cents, we would, using the above key, 
write bk. The following keys are used in some of our 
best business houses: 


HANDY GIRLS 
BLACK HORSE 
THUMB SCR EW 
®ULcCK PAYER 
CUMBE RLAND 
SILVE RTOWN 
VANDE RBILT 
SPECU LATOR 
IMPOR TANCE 
CHARL ESTON 


MISCELLANEOUS. 103 


TO FIND HER AGE. 


316. Have Your Girl write the day of the month 
on which she was born, annex two ciphers, add the num- 
ber of the month in which she was born, multiply by 2, 
add 5, annex two ciphers, divide by 2, add her age in 
years, and subtract 250, and tell you the result. 


317. 1. Suppose the person is 25 years old and was 
born October 26. 
PROCESS, 


2600 
10 


2610 
2 


3220 
5 


2) 522500. 
261250 
25 
261275 
250 


261025 


ExpianaTion:—The two right hand figures will show 
her age in years; the next two the month, and the re- 
mainder, the day of the month on which she was born. 


MAGIC WORDS. 


318. The Magie Words are composed of ten 
letters, each occuring twice. They are here appended 


104 MISCELLANEOUS. 


as a matter of interest and discipline for the pupil. 


ae) 


DE TI D | 
|) NOMEN | 
| 


ExpLaNnation:—Take twenty small articles of any kind 
and divide them into couplets. Let each person present 
choose a couplet. Then place the pieces in four rows, 
like the words, being careful to place the pieces of each 
couplet upon corresponding letters of the imaginary 
words. It will now be an easy matter to pick out any 
couplet, when you know the row or rows in which its 
members are found. 


MAGIC SQUARES. 


319. A Magic Square is a square divided into a 
number of smaller squares in which are inserted numbers, 
generally in arithmetical progression, in such an order 
that each line, whether added vertically, horizontally, or 
diagonally shall amount to the same sum. 


320. 1. Make a magic square, using the nine digits. 


PROCESS. 
2 | 716 
Se ea os | 
4|3|8| 


Expianation:-—Write the numbers according to the 


MISCELLANEOUS. 105 


accompanying diagrams, which explain themselves. 
In constructing a square of nine figures, divide the num- 
ber to which you wish the square to add by 3, and it 
will give the average or middle number. By careful 
study and a little practice the student will have but 
little difficulty in making a square which will add to any 
number he wishes. 


ARRANGEMENT, KEY, 
1234 em 
abed se Sia alge 
«5 A| x | a | 
ABCD eis a 
98S 76 djec |B 


2. Make a magic square, using all the numbers 
from | to 16 inclusive. 


PROCESS, KEY. 
| 2/15|14| 4 a|B/C/| da) 
12, 6| 7| 9 E|ft|/¢|H) 
8 10 1b 5 [h|@| Fle 
/ 7 : = 
13 3) 2,16 Dic |b/|A 
ARRANGEMENT. 
12345678 
a bedefgih 
ABCDEFGH 
1615 1413 121110 9 


Nore:—Although this puzzle has occupied the attention 
of many celebrated mathematicians from the earliest 


106004 MISCELLANEOUS. 


times down to the present day, the above simple method 
is now published for the first time. The magic square is 
of but little use to the ordinary business man. Itis, how- 
ever, a very interesting diversion and well worth investi- 
gation by all who appreciate the wonderful. Any set 
of numbers increasing or decreasing by a common 
difference, may be arranged according to the above 
formulas. 


3. Make a magic square for the year 1888, using all 
the odd numbers from 457 to 487 inclusive. 

4, Using nine numbers, construct a magic square 
which will add to my age, 21 years. 

5. Construct a magic square for the year 1890, using 
nine numbers. 

6. Construct a magic square which will add to 100, 
using all the even numbers from 10 to 40 inclusive. 


PROMISCUOUS PROBLEMS. 


321. 1. Whatisathirdandhalfathirdof10? 5. 


2. Express 100 by using the figure 9 four times. 


998; 
3. How many hands high is a horse which measures 
5 feet? Le 
4, How many steps of 2 ft. 8 in. each will a man take 
in walking 2 miles? 3960. 
5. What is the product of the complement of 80 
and the reciprocal of 35? 500. 
6. How many tons of hay in a stack 20 feet long, 
15 feet high, and 14 feet wide? 3. 
7. The sum of four equal numbers less 75 is 455. 
What is one of the numbers? 132. 


8. The sum of three equal numbers divided by 4 is 


81. What is one of the numbers? 108. 
9. If 2 cats can kill 2 rats in 2 minutes, how many 

eats can kill 100 rats in 100 minutes? 2. 
10. Multiply the sum of 2 and 2} by their difference 

and extract the square root of the product. 1}. 


11. Sold wheat at 63 cents and lost 123 per cent. At 
what should I have sold to gain 84 per cent? 78¢. 


108 MISCELLANEOUS. 


12. How many feet, board measure, in 6 pieces 4 x 4 
at one end, 6 x 6 at the other, and 18 feet long? 234. 


13. Ifa hen and a half lay an egg and a half in a 
day and a half, how many eggs will one hen lay in 30 
days? 20. 

14. Bought bonds at 75 and sold them the same 
day at 102 What was my per cent of gain on the in- 
vestment? 36. 


15. If you buy a saddle for 12 dollars, sell it for 15 
dollars, and buy it back for 9 dollars, what per cent do 
you make? 50. 


16. How many bushels of wheat will a bin hold that 
is 183 feet 6 inches long, 8 feet 4 inches wide, and 6 feet 
8 inches deep? 600. 


17. A man was married at the age of 25. If he lives 
16 years longer he will have been married 47 years. 
What is his age? 56. 


18. When wood is selling at $4.50 per eord, how 
much must I pay for a range 73 feet long, 43 feet wide, 
and 13 feet high? $2.08. 


19. If a cow and a calf eat a pumpkin and a half ina ~ 
day and a half, how many pumpkins will they eatin a 
month and a half? 45, 


20. <A clerk spends 3 of his salary for board, and 3 
of the remainder for clothes, am, saves 75 dollars a year. 
What is his salary? 2 SO h Ds 


21. Ifa sounding line is 100 Pr oehs long and all 
but 24 fathoms and 3 feet is let out, what is the depth 
of the water, in yards? | 151. 


22. What must be the asking price of cloth, costing 
$3.92 a yard, that I may deduct 123 per cent from it and 
still gain 12} per cent? $5.04. 


MISCELLANEOUS. 109 


23. How many shingles will be required to cover a 
house 34 feet long, the rafters 12 feet long, showing 
4 inches to the weather? 7344. 


24. A contractor having | mile of road to grade, has 
completed 319 rods, 5 yards, 1 foot, and 6 inches. How 
much more has he to grade? 


25. Ifa pole 10 feet long casts a shadow 16 feet long, 
what is the height of a steeple which will cast a shadow 
200 feet long at the same time? 125 ft. 


26. A man being asked how many sheep he had, re- 
plied: “If I had 3 times as many as I have and 5 more, 
I would have: 260.” How many had he? 85. 


27. A giant being asked his height replied: “If from 
~ 4 my height, in inches, 4 be subtracted, 4 of the remain- 
der will be 4.” What was his height? 6 ft. 8 in. 


28. A grocer claimed to sell sugar at cost, but used a 
false pound in selling which contained but 14 ounces. 
What per cent did he gain by the cheat? 144, 


29. A snail, at the bottom of a 20 foot well, climbs 
8 feet aday and slips back 4 feet at night. At that 
rate, how many days will it be in reaching the top? 4. 


30. A fence five boards high is built around a square 
field containing 10 acres; each board is 6 inches wide. 
What will the lumber cost at 3 dollars per hundred? 

$198. 


31. <A can doa piece of work in 30 days, B in 40 days. 
After both werk 5 days, A leaves. How many more 
days should A work, that the work may occupy but 20 
days? 10. 


32. The distance between St. Louis and Kansas City 
is 283 miles. A postal clerk makes the round trip be- 
tween these places, twice a week. How many miles will 
he travel in a year? 58864. 


110 MISCELLANEOUS. 


83. <A flag pole is 63 feet high. How many feet above 
the ground must it be broken in order that the upper 
part, clinging to the stump, may touch the ground 21 
feet from the base? 28. 


34. A jockey bought a horse for 90 dollars, sold him 
for 100 dollars, bought him back for 95 dollars, and 
sold him again for 100 dollars. How much did he make 
by both transactions? $15. 


39. A grocer claimed to sell molasses at cost, but used 
a false gallon measure in selling, which contained 3 
quarts and 1} pints. On a sale of 240 dollars, what did 
he gain by the cheat? $16. 


36. A log 20 feet long, 18 inches in diameter at one 
end and 6 inches at the other, starts rolling on level 
ground. How many revolutions will it make in rolling 
around to the starting point? 40. 


37. A and B together had 640 acres of land. A sold 
B 100 acres and afterwards bought 150 acres of B, and 
then sold C 160 acres, so that he had 60 acres less than 
B. How many acres had each? 320. 


38. Bought a horse, carriage and harness for 288 
dollars. The horse cost 5 times as much as the harness, 
and the carriage cost half as much as both horse and 
harness. What was the price of the horse? $160. 


39. A pile of wood is 114 feet 8 inches long, 4 feet 
wide, and 6 feet and 3 inches high. What must be paid 
for sawing and splitting, allowing 1 dollar per cord for 
sawing and 50 cents per cord for splitting? $33.59.+ 


40. A and B having started at the same time to travel 
toward each other, met in 5 hours. B traveled 3 miles 
an hour faster than A, and both together traveled 55 
miles. How many miles per hour did each travel? 

A 4, B7. 


MISCELLANEOUS. 111 


41. ‘Two boats, leaving places 143 miles apart, sail 
toward each other, one at the rate of 15 miles an hour 
and the-other at the rate of 18 miles an hour. When 
they meet, how many miles has each traveled? 

j 65, 78. 

42. Sold-a house for 720 dollars, taking a note draw- 
ing simple interest at 7 per cent. Not needing the 
money, I did not collect the note until the end of two 
years 5 months and 27 days. What amount did I col- 
_ leet? $845.58. 

43. What must I pay for 1} gross of clothes pins at 
4 cents a dozen, a quintal of fish at 2 cents a pound, 
a half barrel of pork at 7 cents a pound, a half ream of 
paper at 12 cents a quire, anda score of knives at 40 cents 
apiece? $18.92. 

44. A man was hired fora year for 100 dollars and a 
suit of clothes; but at the end of 8months he left, and 
received the suit and 60 dollars in money as full com- 
pensation for the time he had worked, What was the 
value of the suit? $20. 

45. James and John agree to mow a field of grass in 
11 days for 62 dollars. James claims that he can do 
* of the work in the required time, and John claims that 
he can do 4 of the work in the required time. How 
much should John receive? $32. 

46. A boy was engaged to work 30 days on these con- 
ditions: For every day he worked he should have one 
dollar and his board, and for every day he was idle he 
should forfeit 25 cents for damage and board. At 
the end of the month he received 20 dollars. How 
- many days did he work? 22. 

47. A bought 5 peaches for 5 cents, and B 9 peaches 
for 9 cents. They were joined by C, and each one ate 
an equal share. When C left he gave them 14 cents to 
pay for his share, but they quarreled about the division of 
it. How should this amount be divided? <A 1, B 13. 


FIZS MISCELLANEOUS. 


48, A thief bought a coat for 5 dollars and tendered 
a 50 dollar bill in payment. The merchant could not 
make the change, but took the bill to a banker, who 
changed it for him. After the thief had gone, the bank- 
er discovered that the bill was counterfeit. He at once 
demanded good money from the merchant, who was | 
obliged to return him 50 dollars. How much did the 
merchant lose? Coat and $5. 


49, Three travellers met on the plains of Arabia; and 
two of them brought their provisions with them; but the 
. third, not having provided any, proposed to the others 
that they should eat together, and he would pay the value 
of his portion, This being agreed to, A produced 3 
loaves, and B 5 loaves, all of which the travellers ate to- 
gether, and C paid 8 pieces of money as the value of his ~ 
share, with which the others were satisfied, but could not 
agree about the division of it. Upon this, the matter was 
referred to a judge, who decided impartially. How 

many pieces did each receive? ASE aise 


